Analyzer, analysis method, and analysis program

ABSTRACT

The present invention enables calculation of a solution of a non-self-adjoint problem represented by simultaneous differential equations. An analysis device includes: a setting unit that sets an original differential operator of an analysis object and a boundary condition of variables; an adjoint boundary condition calculation unit that calculates an adjoint boundary condition from the boundary condition; and a non-self-adjoint calculation unit that calculates a primal differential operator and a dual differential operator from the original differential operator, and determines a primal eigenfunction and a dual eigenfunction by using primal simultaneous differential equations and dual simultaneous differential equations, as well as the boundary condition and the adjoint boundary condition, thereby calculating a solution of simultaneous differential equations.

TECHNICAL FIELD

The present invention relates to a technique of representing a static or kinetic equilibrium of an analysis object such as a continuum by a differential equation and calculating a solution corresponding to a boundary condition, thereby analyzing a static or kinetic equilibrium or a state of the continuum.

BACKGROUND ART

Usually a static or kinetic equilibrium of a continuum is described by simultaneous partial differential equations, and the purpose thereof is to obtain a solution corresponding to various types of boundary conditions. In the field of engineering, a self-adjoint problem is dealt with often, and a variety of methods suitable for the same has been studied. Among these, the eigenfunction method based on Hilbert's expansion theorem is useful, and there are many examples of application of the same. In contrast, there are few studies on a non-self-adjoint problem, one of which is the eigenfunction method based on Schmidt's expansion theorem (see, for example, Non-Patent Documents 1 and 2). Further, Mikhlin (see, for example, Non-Patent Documents 3 and 4) indicates that using this eigenfunction allows the least squares method to give a correct solution. However, since these theorems are in the category of the integral equation theory and boundary conditions are embedded in the kernel function, they are difficult to use as a general-purpose method, in an aspect. Further, they do not take a form of simultaneous equations for solving a plurality of unknown functions. For this reason and others, necessarily there have been a limited number of application examples.

PRIOR ART DOCUMENT Non-Patent Document

-   Non-Patent Document 1: Richard Courant, David Hilbert; joint     translation by Toshiya SAITO and Shigeya MARUYAMA: Methods of     Mathematical Physics, Vol. 1, TokyoTosho Co., Ltd. the first     edition: 1959. -   Non-Patent Document 2: Erhard Schmidt: Zur Theorie der linearen and     nichtlinearen Integralgleichungen. I., Math. Ann. Bd. 63, 1907, pp.     433-476. -   Non-Patent Document 3: Bruce A. Finlayson; joint translation by     Kyuichiro WASHIZU, Yoshiyuki YAMAMOTO, and Tadahiko KAWAI: The     Method of Weighted Residuals and Variational Principles, BAIFUKAN     Co., Ltd., 1974. -   Non-Patent Document 4: S. G. Mikhlin; translated by T. Boddington:     Variational Methods in Mathematical Physics, Pergamon Press, 1964.

SUMMARY OF THE INVENTION Problem to be Solved by the Invention

In the fields of theory of elasticity, vibration science, mechanics of materials, structural mechanics, and the like, as the principle of virtual work is laid at the basis of the theoretical system, contents thereof are biased to a self-adjoint problem, and it seems that even the existence of a non-self-adjoint problem is not recognized. However, there are many non-self-adjoint problems actually. Therefore, it is necessary to recognize ambiguity involved in the principle of virtual work and reconstruct the system of dynamics from a higher viewpoint, to extend the possibility widely and deeply to induce further development.

It is an object of the present invention to provide an analyzer, an analysis method, and an analysis program, characterized in that a solution of a non-self-adjoint problem represented by simultaneous differential equations can be calculated.

Means for Solving the Problem

An analyzer of the present invention is an analyzer for determining a solution of a differential equation of an analysis object by using data representing original differential operators L_(ij) with respect to the analysis object, wherein, when forces acting on the analysis object are assumed to be f_(i), and variations of dual displacements u_(i*) are assumed to be dual variations δu_(i)*, solutions u_(j) are calculated by the following equation:

$\begin{matrix} {{\sum\limits_{i}\; {\int_{S}{{\left( {{\sum\limits_{j}\; {L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}^{*}\ {s}}}} = 0} & \left\lbrack {{Formula}\mspace{14mu} 1} \right\rbrack \end{matrix}$

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 illustrates relationship of an eigenfunction with differential operators or a boundary condition.

FIG. 2 is a functional block diagram illustrating an exemplary configuration of an analyzer of Embodiment 1.

FIG. 3 is a functional block diagram illustrating an exemplary configuration of an analyzer of Embodiment 2.

FIG. 4 is a flowchart showing an exemplary operation of the analyzer shown in FIG. 3.

FIG. 5 is a functional block diagram illustrating an exemplary configuration of an analyzer according to Embodiment 3.

FIG. 6 is a flowchart showing an exemplary operation of the analyzer 10 b shown in FIG. 5.

FIG. 7 is a flowchart showing an exemplary operation of an analyzer according to Embodiment 4.

FIG. 8 illustrates exemplary elements of a 16-nodes quadrilateral.

FIG. 9 illustrates a result of an analysis of a ring under uniform gravity in which “zero displacement and zero surface force” is set as a boundary condition.

FIG. 10 illustrates a result of an analysis of a square under uniform gravity in which “zero displacement and zero surface force” is set as a boundary condition of an upper side of the square.

FIG. 11 illustrates a result of an analysis of in-plane deformation of a square plate having fixed peripheries.

FIG. 12 illustrates a result of an analysis of self-weight deformation of square plate elements in the case where peripheries thereof are fixed, the element being an element for out-of-plane deformation.

FIG. 13A illustrates a model of a beam having a left end under a free boundary condition and fixed boundary condition.

FIG. 13B illustrates a result of calculation of primal eigenfunctions φ₁, φ₂, and φ₃, and dual eigenfunctions φ_(1*), φ_(2*), and φ_(3*) of the beam shown in FIG. 13A.

FIG. 13C illustrates an analytical solution and solutions determined by the eigenfunction method.

FIG. 14 illustrates an analysis result in the case where a virtual boundary of a square is defined in a boundless region of the infinite, and a source is arranged at the center of the square.

FIG. 15 illustrates an analysis result in the case where a virtual boundary of a square is defined in a boundless region of the infinite, and a vortex is arranged at the center of the square.

FIG. 16 illustrates an exemplary spring damper system.

FIG. 17 illustrates an exemplary eigenfunction.

FIG. 18 illustrates comparison between analytical solutions of eq. (525) and results obtained by an eigenfunction method of eq. (494).

FIG. 19 illustrates comparison between analytical solutions of eq. (528) and results obtained by the eigenfunction method of eq. (494).

FIG. 20 illustrates comparison between analytical solutions of eq. (531) and results obtained by the eigenfunction method of eq. (494).

FIG. 21 illustrates an exemplary mass-point spring system.

FIG. 22 illustrates respective eigenfunctions in the case of eq. (610).

FIG. 23 illustrates respective eigenfunctions in the case of eq. (611).

FIG. 24 illustrates comparison between analytical solutions of eq. (625) and results obtained by an eigenfunction method of eq. (578).

FIG. 25 illustrates comparison between analytical solutions of eq. (632) and results obtained by an eigenfunction method of eq. (584).

FIG. 26 illustrates an exemplary static deflection of a string.

FIG. 27 illustrates exemplary eigenfunctions.

FIG. 28 illustrates comparison between analytical solutions of eq. (706) and results obtained by an eigenfunction method of eq. (702).

FIG. 29 illustrates comparison between analytical solutions of eq. (709) and results obtained by the eigenfunction method of eq. (702).

FIG. 30 illustrates comparison between analytical solutions of eq. (713) and results obtained by an eigenfunction method of eq. (704).

FIG. 31 illustrates comparison between analytical solutions of eq. (715) and results obtained by the eigenfunction method of eq. (704).

FIG. 32 illustrates an exemplary static deflection of a beam.

FIG. 33 illustrates exemplary eigenfunctions.

FIG. 34 illustrates exemplary eigenfunctions.

FIG. 35 illustrates comparison between analytical solutions of eq. (784) and results obtained by an eigenfunction method of eq. (780).

FIG. 36 illustrates comparison between analytical solutions of eq. (787) and results obtained by the eigenfunction method of eq. (780).

FIG. 37 illustrates comparison between analytical solutions of eq. (790) and results obtained by an eigenfunction method of eq. (782).

FIG. 38 illustrates comparison between analytical solutions of eq. (793) and results obtained by the eigenfunction method of eq. (782).

FIG. 39 illustrates one solution other than an exemplary solution, among a plurality of solutions.

FIG. 40A illustrates a plurality of exemplary solutions calculated in the case of analysis with respect to a finite element method.

FIG. 40B illustrates a plurality of exemplary solutions calculated in the case of analysis with respect to a finite element method.

FIG. 41 illustrates the node number for a 4-nodes shape function.

FIG. 42 illustrates an analytical solution of deformation and stress of a ring.

FIG. 43 illustrates deformation of a mode SA.

FIG. 44 illustrates deformation of a mode AS.

FIG. 45 illustrates deformation of a mode SS.

FIG. 46 illustrates deformation of a mode AA.

FIG. 47 illustrates No. 1 mode of a ring.

FIG. 48 illustrates No. 2 mode of a ring.

FIG. 49 illustrates No. 3 mode of a ring.

FIG. 50 illustrates No. 4 mode of a ring.

FIG. 51 illustrates deformation and stress distribution in the case where 2 modes are used.

FIG. 52 illustrates deformation and stress distribution in the case where 3 modes are used.

FIG. 53 illustrates deformation and stress distribution in the case where 10 modes are used.

FIG. 54 illustrates deformation and stress distribution in the case where 30 modes are used.

FIG. 55 illustrates a state of a dimensionless velocity of a source.

FIG. 56 illustrates a state of a dimensionless velocity of a vortex.

FIG. 57 illustrates No. 1 mode (m=1,N=1) of a function (1).

FIG. 58 illustrates No. 1 mode (m=1,N=1) of a function (2).

FIG. 59 illustrates No. 1 mode (m=1,N=1) of a function (3).

FIG. 60 illustrates No. 1 mode (m=1,N=1) of a function (4).

FIG. 61 illustrates No. 1 mode (m=1,N=1) of a function (5).

FIG. 62 illustrates No. 1 mode (m=1,N=1) of a function (6).

FIG. 63 illustrates No. 1 mode (m=1,N=1) of a function (7).

FIG. 64 illustrates No. 1 mode (m=1,N=1) of a function (8).

FIG. 65 illustrates a velocity distribution of a source using the eigenfunction method.

FIG. 66 illustrates difference of velocity distribution between analytical solution and eigenfunction method for source.

FIG. 67 illustrates an exemplary configuration of an information processing device.

FIG. 68 illustrates an exemplary configuration of an information processing device.

FIG. 69 is a flowchart illustrating an exemplary boundary condition setting.

FIG. 70 is a flowchart illustrating an exemplary boundary condition setting.

FIG. 71 is a flowchart illustrating an exemplary boundary condition setting.

FIG. 72 is a flowchart illustrating an exemplary analysis processing.

MODE FOR CARRYING OUT THE INVENTION 1. Introduction

Here, a novel eigenfunction method for solving a non-self-adjoint problem represented by simultaneous partial differential equations is shown. Next, the relationship of the present method with the least squares method, the direct variational method, and the energy principle of the theory of elasticity is discussed. Thereafter, the principle of virtual work and the finite element formulation are discussed deeply. Further, formulation of a finite element of a two-dimensional elastic body is discussed.

2. Simultaneous Partial Differential Equations 2.1 Technical Terms and Coinages

Essential terms and coined words relating to the relationship of eigenfunctions with differential operators (differential operators) and boundary conditions are shown in FIG. 1.

A problem to be solved is referred to as a “primal problem”. From an inhomogeneous boundary condition given to the primal problem, a homogeneous boundary condition is settled. There are many problems having the same homogeneous boundary condition, and these problems form an additive group (module) having calculus with a sum of boundary conditions. This is referred to as a “heads group”. In other words, an inhomogeneous boundary condition of the primal problem is included in the heads group as one of the elements. A solution function of the primal problem is referred to as “Primal Solution”, and a variation of the same is referred to as a “primal variation”.

By partial integration of an inner product, adjoint differential operators are settled from original differential operators, and from a condition in which a boundary term is zero, a homogeneous adjoint boundary condition is settled. There are many problems having the same homogeneous adjoint boundary condition, and these form an additive group (module) likewise. This is referred to as a “tails group”. One of the elements is referred to as an “inhomogeneous adjoint boundary condition”, and a problem that gives this is referred to as a “dual problem”. A solution function of a dual problem is referred to as a “dual solution”, and a variation thereof is referred to as a “dual variation”. Via the inner product, the two groups are united like two sides of the same coin.

2.2 Exemplary Self-Adjoint Operator (Static Equilibrium of Elastic Body)

Static equilibrium of a two-dimensional elastic body are, in the orthogonal coordinate system, given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 5} \right\rbrack & \; \\ {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} u_{x}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}u_{y}}} = {{- \frac{1}{G}}b_{x}}} & (1) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}u_{x}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} u_{y}}} = {{- \frac{1}{G}}b_{y}}} & (2) \end{matrix}$

where u_(x), u_(y) represent displacements in the x and y directions, respectively, which are determined as primal solutions by solving the equations. b_(x) and b_(y) are body forces in the x and y directions per unit volume, and G represents a modulus of rigidity. Let a Poisson's ratio (Poisson's ratio) be ν. Then, a constant μ is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 6} \right\rbrack & \; \\ {{\mu \equiv {\frac{1 + v}{1 - v}\left( {{Plane}\mspace{14mu} {Stress}} \right)}},{\mu \equiv {\frac{1}{1 - {2\; v}}\left( {{Plane}\mspace{14mu} {Strain}} \right)}}} & (3) \end{matrix}$

Regarding a plane stress and a plane strain state, the displacement-strain relationship is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 7} \right\rbrack & \; \\ {ɛ_{x} \equiv {\frac{\partial}{\partial x}u_{x}}} & (4) \\ {ɛ_{y} \equiv {\frac{\partial}{\partial y}u_{y}}} & (5) \\ {\gamma_{xy} \equiv {{\frac{\partial}{\partial y}u_{x}} + {\frac{\partial}{\partial x}u_{y}}}} & (6) \end{matrix}$

and the stress-strain relationship is given as:

[Formula 8]

σ_(x) =G{(μ+1)ε_(x)+(μ−1)ε_(y})  (7)

σ_(y) =G{(μ−1)ε_(x)+(μ+1)ε_(y})  (8)

τ_(xy) =Gγ _(xy)  (9)

Therefore, stress components are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 9} \right\rbrack & \; \\ {\sigma_{x} \equiv {G\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{x}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial u_{y}}{\partial y}}} \right\}}} & (10) \\ {\sigma_{y} \equiv {G\left\{ {{\left( {\mu - 1} \right)\frac{\partial u_{x}}{\partial x}} + {\left( {\mu + 1} \right)\frac{\partial u_{y}}{\partial y}}} \right\}}} & (11) \\ {\tau_{xy} = {\tau_{yx} \equiv {G\left( {\frac{\partial u_{y}}{\partial x} + \frac{\partial u_{x}}{\partial y}} \right)}}} & (12) \end{matrix}$

Let components of an outward unit normal vector of a boundary surface be n_(x) and n_(y), and then, surface forces p_(x) and p_(y) are, according to the Cauchy's formula (Cauchy's formula), given as:

[Formula 10]

p _(x) ≡n _(x)σ_(x) +n _(y)τ_(yx)  (13)

p _(y) ≡n _(x)τ_(xy) +n _(y)σ_(y)  (14)

This results in that by the equations in which the constant μ is used as is the case with eqs. (1) to (14), the plane strain state and the plane strain state can be dealt with uniformly. A differential operator of eqs. (1) and (2) is referred to as an “original differential operator L_(ij)”, and is defined as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 11} \right\rbrack & \; \\ {\begin{bmatrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{bmatrix} \equiv \begin{bmatrix} {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} & {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \\ {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} & {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \end{bmatrix}} & (15) \end{matrix}$

Then, by partial integration to be described in the next chapter, an adjoint differential operator L_(ij)* is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 12} \right\rbrack & \; \\ {\begin{bmatrix} L_{11}^{*} & L_{21}^{*} \\ L_{12}^{*} & L_{22}^{*} \end{bmatrix} \equiv \begin{bmatrix} {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} & {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \\ {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} & {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \end{bmatrix}} & (16) \end{matrix}$

An set operator that satisfies

L _(ij) *=L _(ij)  (17)

as is the case with the foregoing example, is referred to as a “self-adjoint differential operator”. If the every component of adjoint differential operator L_(ji)* is equal to that of the original differential operator Lij in which indexes i, j are inverted with respect to the indexes j, i of Lji*, Lij is self-adjoint differential operator.

2.3 Exemplary Non-Self-Adjoint Operator (Potential Flow)

If trying to solve a problem of two-dimensional potential flow only with speed, without using velocity potential, we obtain, in the orthogonal coordinate system, simultaneous partial differential equations that are composed of an equation of continuity (18) and a vortex-free condition (19):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 13} \right\rbrack & \; \\ {{\frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y}} = 0} & (18) \\ {{\frac{\partial u_{y}}{\partial x} - \frac{\partial u_{x}}{\partial y}} = 0} & (19) \end{matrix}$

where u_(x), u_(y) represent velocities in the x and y directions, respectively, which are determined as primal solutions by solving the equation. Differential operators of eqs. (18) and (19) are referred to as “original differential operators L_(ij)”, and are defined as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 14} \right\rbrack & \; \\ {\begin{bmatrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{bmatrix} \equiv \begin{bmatrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \\ {- \frac{\partial}{\partial y}} & \frac{\partial}{\partial x} \end{bmatrix}} & (20) \end{matrix}$

Then, by partial integration that is to be described in the next chapter, adjoint differential operators L_(ij)* are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 15} \right\rbrack & \; \\ {\begin{bmatrix} L_{11}^{*} & L_{21}^{*} \\ L_{12}^{*} & L_{22}^{*} \end{bmatrix} \equiv \begin{bmatrix} {- \frac{\partial}{\partial x}} & \frac{\partial}{\partial y} \\ {- \frac{\partial}{\partial y}} & {- \frac{\partial}{\partial x}} \end{bmatrix}} & (21) \end{matrix}$

A set of operators that satisfies

L _(ji) *≠L _(ij),  (22)

as is the case with the foregoing example, is referred to as “non-self-adjoint differential operators”. If at least one component of adjoint differential operator Lji* is not equal to that of the original differential operator Lij in which indexes i, j are inverted with respect to the indexes j, i of Lji*, Lij is non-self-adjoint differential operator.

3. Primal Problem and Dual Problem 3.1 Homogenization of Boundary Condition

Hereinafter, the exemplary elastic body mentioned in the previous chapter is focused. Displacements u_(x) and u_(y) determined as primal solutions are referred to as “primal displacements”, and are written as “u₁” and “u₂”. Let the right sides of the equations be written as “f₁” and “f₂”, and then, simultaneous partial differential equations (1) and (2) are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 16} \right\rbrack & \; \\ {{\sum\limits_{j}\; {L_{ij}u_{j}}} = f_{i}} & (23) \end{matrix}$

An index B is added to a term that satisfies an inhomogeneous boundary condition so as to let the term be U_(Bj), and an index H is added to a term that satisfies a homogeneous boundary condition so as to let the term be u_(Hj). A primal displacement u_(j) is expressed by a sum of these, which is given as:

[Formula 17]

u _(j) ≡u _(Bj) +u _(Hj)  (24)

According to eqs. (10) to (14) and this equation, the stress and the surface force are divided into terms formed with u_(Bj) or u_(Hj), and are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 18} \right\rbrack & \; \\ {\sigma_{x} \equiv {\sigma_{Bx} + \sigma_{Hx}}} & (25) \\ {\sigma_{Bx} \equiv {G\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{B\; 1}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial u_{B\; 2}}{\partial y}}} \right\}}} & (26) \\ {\sigma_{Hx} \equiv {G\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{H\; 1}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial u_{H\; 2}}{\partial y}}} \right\}}} & (27) \\ {\sigma_{y} \equiv {\sigma_{By} + \sigma_{Hy}}} & (28) \\ {\sigma_{By} \equiv {G\left\{ {{\left( {\mu - 1} \right)\frac{\partial u_{B\; 1}}{\partial x}} + {\left( {\mu + 1} \right)\frac{\partial u_{B\; 2}}{\partial y}}} \right\}}} & (29) \\ {\sigma_{Hy} \equiv {G\left\{ {{\left( {\mu - 1} \right)\frac{\partial u_{H\; 1}}{\partial x}} + {\left( {\mu + 1} \right)\frac{\partial u_{H\; 2}}{\partial y}}} \right\}}} & (30) \\ {\tau_{xy} \equiv {\tau_{Bxy} + \tau_{Hxy}}} & (31) \\ {\tau_{Bxy} = {\tau_{Byx} \equiv {G\left( {\frac{\partial u_{B\; 2}}{\partial x} + \frac{\partial u_{B\; 1}}{\partial y}} \right)}}} & (32) \\ {\tau_{Hxy} = {\tau_{Hyx} \equiv {G\left( {\frac{\partial u_{H2}}{\partial x} + \frac{\partial u_{H\; 1}}{\partial y}} \right)}}} & (33) \\ {p_{x} \equiv p_{1} \equiv {p_{Bx} + p_{Hx}}} & (34) \\ {p_{Bx} \equiv p_{B\; 1} \equiv {{n_{x}\sigma_{Bx}} + {n_{y}\tau_{Byx}}}} & (35) \\ {p_{Hx} \equiv p_{H\; 1} \equiv {{n_{x}\sigma_{Hx}} + {n_{y}\tau_{Hyx}}}} & (36) \\ {p_{y} \equiv p_{2} \equiv {p_{By} + p_{Hy}}} & (37) \\ {p_{By} \equiv p_{B\; 2} \equiv {{n_{x}\tau_{Bxy}} + {n_{y}\sigma_{By}}}} & (38) \\ {p_{Hy} \equiv p_{H\; 2} \equiv {{n_{x}\tau_{Hxy}} + {n_{y}\sigma_{Hy}}}} & (39) \end{matrix}$

u_(Hj) and p_(Hj) are unknown functions that satisfy a homogeneous boundary condition, but since p_(Hj) is expressed with u_(Hj), a function that is finally unknown is only u_(Hj). Similarly, u_(Bj) and p_(Bj) are known functions that satisfy an inhomogeneous boundary condition, but since p_(Bj) is expressed with u_(Bj), a function that is finally known is only u_(Bj). u_(Bj) does not necessarily satisfy eq. (23), and when it is in the form of eq. (24), it satisfies eq. (23). Substituting eq. (24) into eq. (23), we obtain a simultaneous partial differential equation represented by a homogeneous boundary condition, which is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 19} \right\rbrack & \; \\ {{\sum\limits_{j}\; {L_{ij}u_{Hj}}} = f_{Hi}} & (40) \end{matrix}$

where the external force term f_(Hi) is given as:

$\begin{matrix} {f_{Hi} \equiv {f_{i} - {\sum\limits_{j}\; {L_{ij}u_{Bj}}}}} & (41) \end{matrix}$

The foregoing procedure is a conventional practice for solving differential equations, and the problem is changed to a problem of finding an unknown function u_(Hj) depending on a known function u_(Bj).

3.2 Dual Displacement

A displacement that is to be a dual solution is referred to as a “dual displacement u_(r)”. Similarly to the case described in the previous section, the dual displacement u_(j)* is given by the sum of the term u_(Bj)* satisfying the inhomogeneous adjoint boundary condition and the term u_(Hj)* satisfying the homogeneous adjoint boundary condition, which is as follows:

[Formula 20]

u _(j) *≡u _(Bj) *+u _(Hj)*  (42)

According to eqs. (10) to (14) and this equation, the stress and the surface force are also divided into the terms formed with u_(Bj)* or u_(Hj)*, and are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 21} \right\rbrack & \; \\ {\sigma_{x}^{*} \equiv {\sigma_{Bx}^{*} + \sigma_{Hx}^{*}}} & (43) \\ {\sigma_{Bx}^{*} \equiv {G\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{B\; 1}^{*}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial u_{B\; 2}^{*}}{\partial y}}} \right\}}} & (44) \\ {\sigma_{Hx}^{*} \equiv {G\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{H\; 1}^{*}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial u_{H\; 2}^{*}}{\partial y}}} \right\}}} & (45) \\ {\sigma_{y}^{*} \equiv {\sigma_{By}^{*} + \sigma_{Hy}^{*}}} & (46) \\ {\sigma_{By}^{*} \equiv {G\left\{ {{\left( {\mu - 1} \right)\frac{\partial u_{B\; 1}^{*}}{\partial x}} + {\left( {\mu + 1} \right)\frac{\partial u_{B\; 2}^{*}}{\partial y}}} \right\}}} & (47) \\ {\sigma_{Hy}^{*} \equiv {G\left\{ {{\left( {\mu - 1} \right)\frac{\partial u_{H\; 1}^{*}}{\partial x}} + {\left( {\mu + 1} \right)\frac{\partial u_{H\; 2}^{*}}{\partial y}}} \right\}}} & (48) \\ {\tau_{xy}^{*} \equiv {\tau_{Bxy}^{*} + \tau_{Hxy}^{*}}} & (49) \\ {\tau_{Bxy}^{*} = {\tau_{Byx}^{*} \equiv {G\left( {\frac{\partial u_{B\; 2}^{*}}{\partial x} + \frac{\partial u_{B\; 1}^{*}}{\partial y}} \right)}}} & (50) \\ {\tau_{Hxy}^{*} = {\tau_{Hyx}^{*} \equiv {G\left( {\frac{\partial u_{H2}^{*}}{\partial x} + \frac{\partial u_{H\; 1}^{*}}{\partial y}} \right)}}} & (51) \\ {p_{x}^{*} \equiv p_{1}^{*} \equiv {p_{Bx}^{*} + p_{Hx}^{*}}} & (52) \\ {p_{Bx}^{*} \equiv p_{B\; 1}^{*} \equiv {{n_{x}\sigma_{Bx}^{*}} + {n_{y}\tau_{Byx}^{*}}}} & (53) \\ {p_{Hx}^{*} \equiv p_{H\; 1}^{*} \equiv {{n_{x}\sigma_{Hx}^{*}} + {n_{y}\tau_{Hyx}^{*}}}} & (54) \\ {p_{y}^{*} \equiv p_{2}^{*} \equiv {p_{By}^{*} + p_{Hy}^{*}}} & (55) \\ {p_{By}^{*} \equiv p_{B\; 2}^{*} \equiv {{n_{x}\tau_{Bxy}^{*}} + {n_{y}\sigma_{By}^{*}}}} & (56) \\ {p_{Hy}^{*} \equiv p_{H\; 2}^{*} \equiv {{n_{x}\tau_{Hxy}^{*}} + {n_{y}\sigma_{Hy}^{*}}}} & (57) \end{matrix}$

u_(Hj)* and p_(Hj)* are unknown functions that satisfy a homogeneous adjoint boundary condition, but since pH,* is expressed with u_(Hj)*, a function that is finally unknown is only u_(Hj)*. Similarly, u_(Bj)* and p_(Bj)* are known functions that satisfy an inhomogeneous adjoint boundary condition, but since p_(Bj)* is expressed with u_(Bj)*, a function that is finally known is only u_(Bj)*.

3.3 Partial Integration

Let the boundary surface be c, and let the inside of the body be s. A sum of integration obtained by multiplying eq. (23) by the dual displacement u_(i)*, that is, an inner product is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 22} \right\rbrack & \; \\ {{\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{L_{ij}{u_{j} \cdot u_{i}^{*}}\ {s}}}}} = {\sum\limits_{i}\; {\int_{S}{{f_{i} \cdot u_{i}^{*}}\ {s}}}}} & (58) \end{matrix}$

Partial integration of the left side of the equation gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 23} \right\rbrack & \; \\ {{\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{L_{ij}{u_{j} \cdot u_{i}^{*}}\ {s}}}}} = {R + {\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{{u_{j} \cdot L_{ij}^{*}}u_{i}^{*}\ {s}}}}}}} & (59) \end{matrix}$

where R represents a boundary term, which, in the case of the elastic body described in the previous chapter, is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 24} \right\rbrack & \; \\ {R \equiv {\frac{1}{G}{\sum\limits_{i}\; {\int_{C}{\left( {{p_{i}u_{i}^{*}} - {p_{i}^{*}u_{i}}} \right)\ {c}}}}}} & (60) \end{matrix}$

An inner product of the homogenized simultaneous partial differential equation (40) and the function u_(Hj)* is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 25} \right\rbrack & \; \\ {{\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{L_{ij}{u_{Hj} \cdot u_{Hi}^{*}}\ {s}}}}} = {\sum\limits_{i}\; {\int_{S}{{f_{Hi} \cdot u_{Hi}^{*}}\ {s}}}}} & (61) \end{matrix}$

Partial integration of the left side of the equation gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 26} \right\rbrack & \; \\ {{\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{L_{ij}{u_{Hj} \cdot u_{Hi}^{*}}\ {s}}}}} = {R_{H} + {\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{{u_{Hj} \cdot L_{ij}^{*}}u_{Hi}^{*}\ {s}}}}}}} & (62) \end{matrix}$

where R_(H) represents a boundary term, which is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 27} \right\rbrack & \; \\ {R_{H} \equiv {\frac{1}{G}{\sum\limits_{i}\; {\int_{C}{\left( {{p_{Hi}u_{Hi}^{*}} - {p_{Hi}^{*}u_{Hi}}} \right)\ {c}}}}}} & (63) \end{matrix}$

In the case where both of the inhomogeneous boundary condition and the inhomogeneous adjoint boundary condition are zero, eq. (60) and eq. (63) coincide with each other. In the state where homogeneous boundary condition is imposed on u_(Hi) and p_(Hi), the condition that the boundary term R_(H) is zero, that is, the homogeneous adjoint boundary condition is imposed on u_(Hi)* and p_(Hi)*. As a result, eq. (62) is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 28} \right\rbrack & \; \\ {{\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{L_{ij}{u_{Hj} \cdot u_{Hi}^{*}}\ {s}}}}} = {\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{{u_{Hi} \cdot L_{ji}^{*}}u_{Hj}^{*}\ {s}}}}}} & (64) \end{matrix}$

In the case where homogeneous boundary condition and homogeneous adjoint boundary condition coincide with each other, these conditions are referred to as “Self-Adjoint Boundary Condition”. In the case where homogeneous boundary condition and homogeneous adjoint boundary condition do not coincide with each other, these conditions are referred to as “Non-Self-Adjoint Boundary Condition”. A problem in which both of a differential operator and a boundary condition are self-adjoint, is referred to as “a self-adjoint problem”. A problem in which at least one of a differential operator and a boundary condition is non-self-adjoint, is referred to as “a Non-Self-Adjoint Problem”.

3.4 Relationship Between Primal Problem and Dual Problem

Focusing on operators of the right sides of eqs. (59) and (64), we define a simultaneous partial differential equation of the dual problem as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 29} \right\rbrack & \; \\ {{\sum\limits_{j}\; {L_{ji}^{*}u_{j}^{*}}} = f_{i}^{*}} & (65) \end{matrix}$

where f_(i)* in the right side is an external force of the dual problem. Substituting eq. (42) into this equation gives the following homogenized simultaneous partial differential equation of the dual problem:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 30} \right\rbrack & \; \\ {{\sum\limits_{j}\; {L_{ji}^{*}u_{Hj}^{*}}} = f_{Hi}^{*}} & (66) \end{matrix}$

where the external force term f*_(Hi) is given as:

$\begin{matrix} {f_{Hi}^{*} \equiv {f_{i}^{*} - {\sum\limits_{j}\; {L_{ji}^{*}u_{Bj}^{*}}}}} & (67) \end{matrix}$

Substituting eqs. (40) and (66) into eq. (64) gives the following equation:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 31} \right\rbrack & \; \\ {{\sum\limits_{i}{\int_{S}^{\;}{{f_{Hi} \cdot u_{Hi}^{*}}\ {s}}}} = {\sum\limits_{i}{\int_{S}^{\;}{{u_{Hi} \cdot f_{Hi}^{*}}\ {s}}}}} & (68) \end{matrix}$

This represents relationship established between solution functions u_(Hi) and u_(Hi)* of the primal and dual problems and the external force terms f_(Hi) and f_(Hi)*.

4. Simultaneous Eigenvalue Problem 4.1 Function Group

Let a function group that satisfies the homogeneous boundary condition be φ_(j), and let the k-th function of this group be written as φ_(jk). Then, the solution function u_(Hj) is given by the sum of the same, which is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 32} \right\rbrack & \; \\ {u_{Hj} \equiv {\sum\limits_{k}{c_{k}\varphi_{jk}}}} & (69) \end{matrix}$

Similarly, let a function group that satisfies the homogeneous adjoint boundary condition be φ_(i)*, and let the k-th function be written as φ_(jk)*. Then, the solution function u_(Hj)* is given by the sum of the same, which is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 33} \right\rbrack & \; \\ {u_{Hj}^{*} \equiv {\sum\limits_{k}{c_{k}^{*}\varphi_{jk}^{*}}}} & (70) \end{matrix}$

where c_(k) and c_(k)* represent k-th coefficients. To obtain a solution function equal “almost everywhere” in a sense of Lebesgue (Lebesgue), the function groups φ_(j) and φ_(j)* should be eigenfunctions. Further, focusing on eq. (68), we find that by composing the external force term f_(Hj) with the function group φ_(j)*, and composing the external force term f_(Hj)* with the function group φ_(j), relationship is provided between the coefficients c_(k) and c_(k)* via the inner product.

4.2 Stress and Surface Force According to Function Groups

Substituting eq. (69) into eqs. (27), (30), and (33) representing stress gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 34} \right\rbrack & \; \\ {\sigma_{Hx} \equiv {G{\sum\limits_{k}{c_{k}\left\{ {{\left( {\mu + 1} \right)\frac{\partial\varphi_{1k}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial\varphi_{2k}}{\partial y}}} \right\}}}}} & (71) \\ {\sigma_{Hy} \equiv {G{\sum\limits_{k}{c_{k}\left\{ {{\left( {\mu - 1} \right)\frac{\partial\varphi_{1k}}{\partial x}} + {\left( {\mu + 1} \right)\frac{\partial\varphi_{2k}}{\partial y}}} \right\}}}}} & (72) \\ {\tau_{Hxy} = {\tau_{Hyx} \equiv {G{\sum\limits_{k}{c_{k}\left( {\frac{\partial\varphi_{2k}}{\partial x} + \frac{\partial\varphi_{1k}}{\partial y}} \right)}}}}} & (73) \end{matrix}$

Then, an index E is added to stress generated from the function group φ_(j), whereby respective stresses are represented as σ_(Ex), σ_(Ey), and τ_(Exy), which are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 35} \right\rbrack & \; \\ {\sigma_{Ex} \equiv {G\left\{ {{\left( {\mu + 1} \right)\frac{\partial\varphi_{1}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial\varphi_{2}}{\partial y}}} \right\}}} & (74) \\ {\sigma_{Ey} \equiv {G\left\{ {{\left( {\mu - 1} \right)\frac{\partial\varphi_{1}}{\partial x}} + {\left( {\mu + 1} \right)\frac{\partial\varphi_{2}}{\partial y}}} \right\}}} & (75) \\ {\tau_{Exy} = {\tau_{Eyx} \equiv {G\left( {\frac{\partial\varphi_{2}}{\partial x} + \frac{\partial\varphi_{1}}{\partial y}} \right)}}} & (76) \end{matrix}$

Surface forces p_(Ex) and p_(Ey) generated from the function group φ_(j) are given as:

[Formula 36]

p _(Ex) ≡p _(E1) ≡n _(x)σ_(Ex) +n _(y)τ_(Eyx)  (77)

p _(Ey) ≡p _(E2) ≡n _(x)τ_(Exy) n _(y)σ_(Ey)  (78)

This is preparation for causing the surface force pE, according to the function group φ_(j) to satisfy the boundary condition.

Similarly, substituting eq. (70) into eqs. (45), (48), and (51) representing stresses gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 37} \right\rbrack & \; \\ {\sigma_{Hx}^{*} \equiv {G{\sum\limits_{k}{c_{k}^{*}\left\{ {{\left( {\mu + 1} \right)\frac{\partial\varphi_{1k}^{*}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial\varphi_{2k}^{*}}{\partial y}}} \right\}}}}} & (79) \\ {\sigma_{Hy}^{*} \equiv {G{\sum\limits_{k}{c_{k}^{*}\left\{ {{\left( {\mu - 1} \right)\frac{\partial\varphi_{1k}^{*}}{\partial x}} + {\left( {\mu + 1} \right)\frac{\partial\varphi_{2k}^{*}}{\partial y}}} \right\}}}}} & (80) \\ {\tau_{Hxy}^{*} = {\tau_{Hyx}^{*} \equiv {G{\sum\limits_{k}{c_{k}^{*}\left( {\frac{\partial\varphi_{2k}^{*}}{\partial x} + \frac{\partial\varphi_{1k}^{*}}{\partial y}} \right)}}}}} & (81) \end{matrix}$

Then, respective stresses generated from the function group φ_(j)* are represented as σ_(Ex)*, σ_(Ey)*, and τ_(Exy)*, which are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 38} \right\rbrack & \; \\ {\sigma_{Ex}^{*} \equiv {G\left\{ {{\left( {\mu + 1} \right)\frac{\partial\varphi_{1}^{*}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial\varphi_{2}^{*}}{\partial y}}} \right\}}} & (82) \\ {\sigma_{Ey}^{*} \equiv {G\left\{ {{\left( {\mu - 1} \right)\frac{\partial\varphi_{1}^{*}}{\partial x}} + {\left( {\mu + 1} \right)\frac{\partial\varphi_{2^{*}}}{\partial y}}} \right\}}} & (83) \\ {\tau_{Exy}^{*} = {\tau_{Eyx}^{*} \equiv {G\left( {\frac{\partial\varphi_{2}^{*}}{\partial x} + \frac{\partial\varphi_{1}^{*}}{\partial y}} \right)}}} & (84) \end{matrix}$

Surface forces p_(Ex)*and p_(Ey)* generated from the function group φ_(j)* are given as:

[Formula 39]

p _(Ex) *≡p _(E1) *≡n _(x)σ_(Ex) *+n _(y)τ_(Eyx)*  (85)

p _(Ey) *≡p _(E2) *≡n _(x)τ_(Exy) *+n _(y)σ_(Ey)*  (86)

This is preparation for causing the surface force p_(Ei)* according to the function group φ_(j)* to satisfy the boundary condition.

4.3 Primal and Dual Simultaneous Differential Equations, and Primal and Dual Simultaneous Eigenvalue Problems

Based on the knowledge described in Section 4.1, focusing on the homogenized differential equation (40), we define primal simultaneous differential equations as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 40} \right\rbrack & \; \\ {{\sum\limits_{j}{L_{ij}\varphi_{j}}} = {\lambda^{*}w_{i}^{*}\varphi_{i}^{*}}} & (87) \end{matrix}$

Further, focusing on the differential equation (66), we define dual simultaneous differential equations as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 41} \right\rbrack & \; \\ {{\sum\limits_{j}{L_{ij}^{*}\varphi_{j}^{*}}} = {\lambda \; w_{i}\varphi_{i}}} & (88) \end{matrix}$

where w_(i), w_(i)* are constants representing weights, and λ, λ* are constants that will become eigenvalues.

An inner product of the primal simultaneous differential equation (87) and φ_(i)* is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 42} \right\rbrack & \; \\ {{\sum\limits_{i}{\sum\limits_{j}{\int_{S}{L_{ij}{\varphi_{j} \cdot \varphi_{i}^{*}}{s}}}}} = {\lambda^{*}{\sum\limits_{i}{\int_{S}{w_{i}^{*}{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}{s}}}}}} & (89) \end{matrix}$

Partial integration of the left side of the equation gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 43} \right\rbrack & \; \\ {{\sum\limits_{i}{\sum\limits_{j}{\int_{S}{L_{ij}{\varphi_{j} \cdot \varphi_{i}^{*}}{s}}}}} = {R_{E} + {\sum\limits_{i}{\sum\limits_{j}{\int_{S}{{\varphi_{j} \cdot L_{ij}^{*}}\varphi_{i}^{*}{s}}}}}}} & (90) \end{matrix}$

where R_(E) represents a boundary term, which is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 44} \right\rbrack & \; \\ {R_{E} \equiv {\frac{1}{G}{\sum\limits_{i}{\int_{C}^{\;}{\left( {{p_{Ei}\varphi_{i}^{*}} - {p_{Ei}^{*}\varphi_{i}}} \right)\ {c}}}}}} & (91) \end{matrix}$

Since the homogeneous boundary condition is imposed on φ_(i) and p_(Ei), and the homogeneous adjoint boundary condition is imposed on φ_(i)* and p_(Ei)* as described in Section 4.1, the boundary term R_(E) becomes zero. Therefore, according to eq. (88), eq. (90) is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 45} \right\rbrack & \; \\ {{\sum\limits_{i}{\sum\limits_{j}{\int_{S}{L_{ij}{\varphi_{j} \cdot \varphi_{i}^{*}}{s}}}}} = {\lambda {\sum\limits_{i}{\int_{S}{{\varphi_{i} \cdot w_{i}}\varphi_{i}{s}}}}}} & (92) \end{matrix}$

The left sides of eqs. (89) and (92) coincide with each other. Therefore, focusing on the right side of the equation, and setting λ and λ* so that they satisfy:

λ*=λ  (93)

we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 46} \right\rbrack & \; \\ {{\sum\limits_{i}{\int_{S}{w_{i}^{*}{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}{s}}}} = {\sum\limits_{i}{\int_{S}{{\varphi_{i} \cdot w_{i}}\varphi_{i}{s}}}}} & (94) \end{matrix}$

In other words, under eq. (93), weighted inner products of the functions φ_(i) and φ_(i)*are equal.

On the other hand, mutually substituting the primal simultaneous differential equation (87) and the dual simultaneous differential equation (88) into each other gives two simultaneous eigenvalue problems as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 47} \right\rbrack & \; \\ {{\sum\limits_{j}{\sum\limits_{k}{L_{ji}^{*}\frac{1}{w_{j}^{*}}L_{jk}\varphi_{k}}}} = {\lambda \; \lambda^{*}w_{i}\varphi_{i}}} & (95) \\ {{\sum\limits_{j}{\sum\limits_{k}{L_{ji}\frac{1}{w_{j}}L_{kj}^{*}\varphi_{k}^{*}}}} = {\lambda \; \lambda^{*}w_{i}^{*}\varphi_{i}^{*}}} & (96) \end{matrix}$

Here, w_(i) and w_(i)* are set so as to satisfy:

w _(i) =w _(i)*  (97)

Then, applying eqs. (93) and (97) to eqs. (95) and (96), we obtain:

$\begin{matrix} {{\sum\limits_{j}^{\;}{\sum\limits_{k}{L_{ij}^{*}\frac{1}{w_{j}}L_{jk}\varphi_{k}}}} = {\lambda^{2}w_{i}\varphi_{i}}} & (98) \\ {{\sum\limits_{j}^{\;}\; {\sum\limits_{k}{L_{ij}\frac{1}{w_{j}}L_{kj}^{*}\varphi_{k}^{*}}}} = {\lambda^{2}w_{i}\varphi_{i}}} & (99) \end{matrix}$

These both are simultaneous eigenvalue problems having an eigenvalue of λ². The eq. (98) is referred to as a “primal simultaneous eigenvalue problem”, and eq. (99) is referred to as a “dual simultaneous eigenvalue problem”. The function φ_(i) is referred to as a “primal eigenfunction”, and  _(i)* is referred to as a “dual eigenfunction”. The operators on the left sides thereof are referred to as a “primal differential operator”, and a “dual differential operator”, respectively. Both of the primal and dual differential operators are self-adjoint differential operators.

By the way, the homogeneous boundary condition is imposed on the function φ_(i), and the homogeneous adjoint boundary condition is imposed on the function φ_(i)*. Since the function φ_(i)* is present on the right side of the primal simultaneous differential equation (87), the function φ_(i) has to satisfy the homogeneous boundary condition, and at the same time, the sum of the derivatives, Σ_(j)L_(ij)φ_(j), has to satisfy the homogeneous adjoint boundary condition. These in combination are referred to as a “primal boundary condition”. Since the function φ_(i) is present on the right side of the dual simultaneous differential equation (88), the function φ_(i)* has to satisfy the homogeneous adjoint boundary condition, and at the same time, the sum of derivatives, Σ_(j)L_(ji)*φ_(j)*, has to satisfy the homogeneous boundary condition. These in combination are referred to as a “dual boundary condition”. Both of the primal and dual boundary conditions are self-adjoint boundary conditions.

φ_(i) of eq. (98) and φ_(i)* of eq. (99), on which these boundary conditions are imposed have own orthogonality and become the basis function in the Hibert space (Hibert space). For example, let the m-th eigenvalue and primal eigenfunction be λ_(m) and φ_(im), respectively, and let the n-th eigenvalue and primal eigenfunction be λ_(n) and φ_(in), respectively. Then, by determining the inner product, we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 48} \right\rbrack} & \; \\ {{\lambda_{n}^{2}{\sum\limits_{i}^{\;}{\int_{S}^{\;}{{\varphi_{im} \cdot w_{i}}\varphi_{in}\ {s}}}}} = {{\sum\limits_{i}{\sum\limits_{j}{\sum\limits_{k}{\int_{S}^{\;}{{\varphi_{im} \cdot L_{ji}^{*}}\frac{1}{w_{j}}L_{jk}\varphi_{kn}\ {s}}}}}} = {{\sum\limits_{i}{\sum\limits_{j}{\sum\limits_{k}{\int_{S}^{\;}{L_{jk}^{*}\frac{1}{w_{j}}L_{ji}{\varphi_{im} \cdot \varphi_{kn}}\ {s}}}}}} = {\lambda_{m}^{2}{\sum\limits_{k}^{\;}{\int_{S}^{\;}{w_{k}{\varphi_{km} \cdot \varphi_{kn}^{*}}\ {s}}}}}}}} & (100) \end{matrix}$

According to the first part and the last part of the foregoing equation, we obtain:

$\begin{matrix} {{\left( {\lambda_{m}^{2} - \lambda_{n}^{2}} \right){\sum\limits_{i}^{\;}{\int_{S}^{\;}{{\varphi_{im} \cdot w_{i}}\varphi_{in}\ {s}}}}} = 0} & (101) \end{matrix}$

Thus, we find that the primal eigenfunction φi has orthogonality. Likewise, let the m-th eigenvalue and dual eigenfunction be φ_(m) and φ_(im)*, respectively, and let the n-th eigenvalue and dual eigenfunction be λ_(n) and φ_(in)*, respectively. Then, by determining the inner product, we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 49} \right\rbrack} & \; \\ {{\lambda_{n}^{2}{\sum\limits_{i}^{\;}{\int_{S}^{\;}{{\varphi_{im}^{*} \cdot w_{i}}\varphi_{in}^{*}\ {s}}}}} = {{\sum\limits_{i}{\sum\limits_{j}{\sum\limits_{k}{\int_{S}^{\;}{{\varphi_{im}^{*} \cdot L_{ji}}\frac{1}{w_{j}}L_{kj}^{*}\varphi_{kn}^{*}\ {s}}}}}} = {{\sum\limits_{i}{\sum\limits_{j}{\sum\limits_{k}{\int_{S}^{\;}{L_{kj}\frac{1}{w_{j}}L_{ij}^{*}{\varphi_{im}^{*} \cdot \varphi_{kn}^{*}}\ {s}}}}}} = {\lambda_{m}^{2}{\sum\limits_{k}^{\;}{\int_{S}^{\;}{w_{k}{\varphi_{km}^{*\;} \cdot \varphi_{kn}^{*}}\ {s}}}}}}}} & (102) \end{matrix}$

According to the first part and the last part of the foregoing equation, we obtain:

$\begin{matrix} {{\left( {\lambda_{m}^{2} - \lambda_{n}^{2}} \right){\sum\limits_{i}^{\;}{\int_{S}^{\;}{w_{i}{\varphi_{im}^{*} \cdot \varphi_{in}^{*}}\ {s}}}}} = 0} & (103) \end{matrix}$

Thus, we find that the dual eigenfunction φ_(i)* also has orthogonality.

Therefore, by taking eqs. (94) and (97) into consideration, we can normalize the functions φ_(i), φ_(i)* so that the functions satisfy:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 50} \right\rbrack & \; \\ \begin{matrix} {{\sum\limits_{i}^{\;}{\int_{S}^{\;}{w_{i}{\varphi_{im}^{*} \cdot \varphi_{in}^{*}}\ {s}}}} = {\sum\limits_{i}{\int_{S}^{\;}{{\varphi_{im} \cdot w_{i}}\varphi_{in}\ {s}}}}} \\ {= \delta_{mn}} \end{matrix} & (104) \end{matrix}$

where δ_(mn) represents the Kronecker's delta (Kronecker's delta).

4.4 Preparation for the Eigenfunction Method

According to eqs. (93) and (97), the primal simultaneous differential equation (87) is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 51} \right\rbrack & \; \\ {{\sum\limits_{j}{L_{ij}\varphi_{j}}} = {\lambda \; w_{i}\varphi_{i}^{*}}} & (105) \end{matrix}$

The dual simultaneous differential equation (88) remains to be as follows, without transformation:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 52} \right\rbrack & \; \\ {{\sum\limits_{j}{L_{ij}^{*}\varphi_{j}^{*}}} = {\lambda \; w_{i}\varphi_{i}}} & (106) \end{matrix}$

The functions φ_(i) and φ_(i)*, which will be used in the eigenfunction method described in the next chapter, are those of a combination that satisfies the primal and dual simultaneous differential equations (105) and (106). It should be noted that a combination in which φ_(i) remains without change and the signs of λ and φ_(i)* are inverted also satisfies the requirements, but this combination is excluded.

In the case where both of the original differential operator L_(ij) and the homogeneous boundary condition are self-adjoint, eqs. (105) and (106) coincide with each other, which form a self-adjoint simultaneous eigenvalue problem given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 53} \right\rbrack & \; \\ {{\sum\limits_{j}{L_{ij}\varphi_{j}}} = {\lambda \; w_{i}\varphi_{i}}} & (107) \end{matrix}$

This equation is a format that represents natural vibration of a Timoshenko beam (Timoshenko beam) or a Mindlin plate (Mindlin plate), and is a key to derive eq. (123) directly from the eigenfunction method.

5. Eigenfunction Method and Least Squares Method

5.1 Eigenfunction method 5.1.1 Primal problem

Substituting eq. (69) into the homogenized differential equation (40) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 54} \right\rbrack & \; \\ {{\sum\limits_{k}{c_{k}{\sum\limits_{j}{L_{ij}\varphi_{jk}}}}} = f_{Hi}} & (108) \end{matrix}$

According to the primal simultaneous differential equation (105), this equation is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 55} \right\rbrack & \; \\ {{\sum\limits_{k}{c_{k}\lambda_{k}w_{i}\varphi_{ik}^{*}}} = f_{Hi}} & (109) \end{matrix}$

where λ_(k) represents the k-th eigenvalue. By determining an inner product with the function φ_(i)* and utilizing the orthogonality, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 56} \right\rbrack & \; \\ {{c_{k}\lambda_{k}{\sum\limits_{i}{\int_{S}^{\;}{w_{i}{\varphi_{ik}^{*} \cdot \varphi_{ik}^{*}}\ {s}}}}} = {\sum\limits_{i}{\int_{S}^{\;}{{f_{Hi} \cdot \varphi_{ik}^{*}}\ {s}}}}} & (110) \end{matrix}$

This allows the coefficient c_(k) to be settled. By normalization of eq. (104), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 57} \right\rbrack & \; \\ {c_{k} = {\frac{1}{\lambda_{k}}{\sum\limits_{i}{\int_{S}^{\;}{{f_{Hi} \cdot \varphi_{ik}^{*}}\ {s}}}}}} & (111) \end{matrix}$

According to this, we find that the coefficient c_(k) is settled by an inner product of the external force term f_(Hi) and the dual eigenfunction φ_(Hi)*. By returning this to eq. (69) whereby the displacement u_(Hj) is determined, and returning the u_(Hj) to eq. (24), we obtain the displacement u_(j). On the other hand, in the case where it is returned to eq. (109), the external force term f_(Hi) is reconstructed, which is useful for verification. In other words, the solution method of eq. (111) is an eigenfunction method in the Hilbert space in which the displacement u_(Hj) is expressed by the primal eigenfunction ch, and the external force term fH, is expressed by the dual eigenfunction φ_(i)*.

5.1.2 Dual Problem

Substituting eq. (70) into the homogenized differential equation (66) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 58} \right\rbrack & \; \\ {{\sum\limits_{k}{c_{k}^{*}{\sum\limits_{j}{L_{ji}^{*}\varphi_{jk}^{*}}}}} = f_{Hi}^{*}} & (112) \end{matrix}$

According to the dual simultaneous differential equation (106), this equation is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 59} \right\rbrack & \; \\ {{\sum\limits_{k}{c_{k}^{*}\lambda_{k}w_{i}\varphi_{ik}}} = f_{Hi}^{*}} & (113) \end{matrix}$

By determining an inner product with the function φ_(i) and utilizing the orthogonality, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 60} \right\rbrack & \; \\ {{c_{k}^{*}\lambda_{k}{\sum\limits_{i}{\int_{S}{w_{i}{\varphi_{ik} \cdot \varphi_{ik}}{s}}}}} = {\sum\limits_{i}{\int_{S}{{f_{Hi}^{*} \cdot \varphi_{ik}}{s}}}}} & (114) \end{matrix}$

This allows the coefficient c_(k)* to be settled. By normalization of eq. (104), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 61} \right\rbrack & \; \\ {c_{k}^{*} = {\frac{1}{\lambda_{k}}{\sum\limits_{i}{\int_{S}{{f_{Hi}^{*} \cdot \varphi_{ik}}{s}}}}}} & (115) \end{matrix}$

According to this, we find that the coefficient c_(k)* is settled by an inner product of the external force term f_(Hi)* and the primal eigenfunction φ_(i). By returning this to eq. (70) whereby the displacement u_(Hj)* is determined, and returning the u_(Hj)* to eq. (42), the displacement u_(j)* is determined. On the other hand, in the case where it is returned to eq. (113), the external force term f_(Hi)* is reconstructed, which is useful for verification. In other words, the solution method of eq. (115) is an eigenfunction method in the Hilbert space in which the displacement u_(Hj)* is expressed by the dual eigenfunction φ_(i)*, and the external force term f_(Hi)* is expressed by the primal eigenfunction φ_(i).

5.1.3 Dual Principle

Here, watching FIG. 1 closely again, we find that in the structure like the two sides of the same coin, the original differential operator corresponds to the material of the coin, the homogeneous boundary condition corresponds to the base for patterns on the heads and tails sides of the coin, and eigenvalues and the primal and dual eigenfunctions existing inside play the leading roles in the solution method.

Further, if starting from the dual problem, the relationship that is similar but in a reverse direction is obtained. Therefore, this shows that the primal and dual eigenfunctions can be used in the reverse state for obtaining the solution. In other words, FIG. 1 demonstrates the dual principle (duality principle) of the simultaneous partial differential equations. The reason why the word “dual” is put in the name lies in the above-described characteristic, i.e., “the structure like the two sides of the same coin, and usability in the reverse state”.

5.2 Direct Variation Method and Least Squares Method

A primal variation δu_(i) and a dual variation δu_(i)* are given as follows, according to eqs. (24) and (42), respectively:

[Formula 62]

δu _(i) ≡δu _(Bi) +δu _(Hi) =δu _(HI)  (116)

δu _(i) *≡δu _(Bi) *+δu _(Hi) *=δu _(HI)*  (117)

Since the term satisfying the inhomogeneous boundary condition is a known function, we use the following in the eqs. (116) and (117):

[Formula 63]

δu _(Bi)=0  (118)

δu _(Bi)*=0  (119)

Consequently, according to eqs. (69), (116), and (70), (117), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 64} \right\rbrack & \; \\ {{\delta \; u_{i}} = {{\delta \; u_{Hi}} \equiv {\sum\limits_{k}{\delta \; c_{k}\varphi_{ik}}}}} & (120) \\ {{\delta \; u_{i}^{*}} = {{\delta \; u_{Hi}^{*}} \equiv {\sum\limits_{k}{\delta \; c_{k}^{*}\varphi_{ik}^{*}}}}} & (121) \end{matrix}$

This is nothing but equations in which the primal variation δu_(i) is expressed with the primal eigenfunction φ_(i), and the dual variation δu_(i)* is expressed with the dual eigenfunction φ_(i)*. Therefore, combining a plurality of procedures for obtaining an inner product with the function φ_(i)*, which is used in the process of obtaining eq. (110), is equivalent to obtaining an inner product with the dual variation δu_(i)* of eq. (121). In other words, the following is established:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 65} \right\rbrack & \; \\ {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij} \cdot u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}^{*}{s}}}} = 0} & (122) \end{matrix}$

Conversely, by creating this equation with respect to a variety of δu_(i)*, and solving simultaneous equations algebraically, the coefficient c_(k) can be determined.

Further, in the case of the self-adjoint problem, the primal eigenfunction φ_(i) and dual eigenfunction φ_(i)* coincide with each other via eq. (107). Therefore, eqs. (120) and (121) become equivalent to each other, and eq. (122) is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 66} \right\rbrack & \; \\ {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}{s}}}} = 0} & (123) \end{matrix}$

On the other hand, according to eqs. (105) and (120), we obtain a sum of differential coefficients of the primal variation δu_(i), which is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 67} \right\rbrack & \; \\ {{\sum\limits_{j}{L_{ij}\delta \; u_{j}}} = {{\sum\limits_{k}{\delta \; c_{k}{\sum\limits_{j}{L_{ij}\varphi_{jk}}}}} = {\sum\limits_{k}{\delta \; c_{k}\lambda_{k}w_{i}\varphi_{ik}^{*}}}}} & (124) \end{matrix}$

With the primal and dual eigenfunctions in which the same weight w is used for the every component, we obtain the following definition:

[Formula 68]

δc _(k) *≡δc _(k)λ_(k) w _(i) , w _(i) =w  (125)

Therefore, according to eqs. (121), (124), and (125), we recognize the dual variation δu_(i)* as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 69} \right\rbrack & \; \\ {{\delta \; u_{i}^{*}} = {{\sum\limits_{j}{L_{ij}\delta \; u_{j}}} = {\delta \; {\sum\limits_{j}{L_{ij}u_{j}}}}}} & (126) \end{matrix}$

By substituting this equation into eq. (122), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 70} \right\rbrack & \; \\ {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; {\sum\limits_{j}{L_{ij}u_{j}{s}}}}}} = 0} & (127) \end{matrix}$

In the case where f_(i) are dealt with as known external forces, what expressed by the equation (127) is equivalent to a case where a functional Π is given as the following and variation thereof is zero:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 71} \right\rbrack & \; \\ {\Pi \equiv {\sum\limits_{i}{\int_{S}{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right)^{2}{s}}}}} & (128) \end{matrix}$

In other words, this shows that the least squares method (least squares method) is the variational principle (variational principle). It should be noted that eqs. (122), (123), and (127) show the direct variational methods (direct variational methods), and eq. (127) is equivalent to eq. (122) in the case of the non-self-adjoint problem, and is equivalent to eq. (123) in the case of the self-adjoint problem.

6. Energy Principle in Theory of Elasticity

A problem for which a body force is specified as known quantity is discussed. According to eq. (128), basically, the least squares method is the variation principle, with respect to all of problems.

Focusing on the boundary term RE of eq. (91), we find that such a condition that

p _(Ei)=0 or φ_(i)=0  (129)

is satisfied on a boundary surface is a self-adjoint boundary condition. In other words, a problem such that either one of the surface force and the displacement is given for each of x and y direction component is a self-adjoint problem. Therefore, in this case, eq. (123) may be used. A total strain energy U per unit thickness is given as:

[Formula 72]

U≡½∫_(S)(σ_(x)ε_(x)+σ_(y)ε_(y)+τ_(xy)γ_(xy))ds  (130)

and its variation δU is given as:

[Formula 73]

δU≡∫ _(S)(σ_(x)δε_(x)+σ_(y)δε_(y)+τ_(xy)δγ_(xy))ds  (131)

By transforming eq. (123) and using this equation, we obtain:

[Formula 74]

δU=∫ _(C)(p _(x) δu _(x) +p _(y) δu _(y))dc+∫ _(S)(b _(x) δu _(x) +b _(y) δu _(y))ds  (132)

Though this is in the same format as that of the principle of virtual work (principle of virtual work), it is eq. (123) itself, and hence, it should be recognized as the direct variational method. Let an area obtained by removing the field having a known displacement from the boundary surface be c_(p). Then, on c_(p), the surface force is a known value. Therefore, what expressed by eq. (132) is equivalent to a case where a functional Π is given as the following and variation thereof is zero:

[Formula 75]

Π≡U−∫ _(Cp)(p _(x) u _(x) +p _(y) u _(y))dc−∫ _(S)(b _(x) u _(x) +b _(y) u _(y))ds  (133)

This is the principle of minimum potential energy (minimum potential energy).

The Clapeyron's theorem (Clapeyron's theorem), which represents the energy conservation law, is given as:

[Formula 76]

U=½∫_(C)(p _(x) u _(x) +p _(y) u _(y))dc+½∫_(S)(b _(x) u _(x) +b _(y) u _(y))ds  (134)

When variations are set herein, we obtain:

[Formula 77]

δU=½∫_(C)δ(p _(x) u _(x) +p _(y) u _(y))dc+½∫_(S)δ(b _(x) u _(x) +b _(y) u _(y))ds  (135)

According to this equation and eq. (132), we obtain:

[Formula 78]

δU=∫ _(C)δ(u _(x) δp _(x) +u _(y) δp _(y))dc+∫ _(S)(u _(x) δb _(x) +u _(y) δb _(y))ds  (136)

Though this is in the same format as that of the principle of complementary virtual work (principle of complementary virtual work), it is a result obtained by applying the energy conservation law to eq. (123), and hence, it should be recognized as another format of the direct variational method. Let an area obtained by removing the field having a known surface force from the boundary surface be Cu. Then, on Cu, the displacement is a known value. Therefore, what expressed by eq. (136) is equivalent to a case where a functional Π is given as the following and variation thereof is zero:

[Formula 79]

Ø≡U−∫ _(Cu)(p _(x) u _(x) +p _(y) u _(y))dc  (137)

This is the principle of minimum work (principle of minimum work).

7. Conclusion

(1) The simultaneous partial differential equation (23) can be solved by the eigenfunction method of eq. (111). According to eq. (128), the least squares method is the variation principle. In the case of the non-self-adjoint problem, eqs. (122) and (127) function as the direct variational method, and in the case of the self-adjoint problem, eqs. (123) and (127) function as the direct variational method.

(2) Since the variation ou, in eq. (123) is formed with self-adjoint eigenfunctions, it is a variation with respect to a displacement that satisfies both of the geometric boundary condition and the mechanical boundary condition. On the other hand, in the principle of virtual work, which is in the same format as that of eq. (123) coincidentally, δu_(i) is referred to as a virtual displacement in “all of functions satisfying the geometric boundary condition”. Since it is not appropriate to use the variation sign δ in a function that does not necessarily satisfy the mechanical boundary condition, if the virtual displacement is written as δ{tilde over ( )}u_(i), then, the principle of virtual work is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 80} \right\rbrack & \; \\ {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \overset{\sim}{\delta}}\; u_{i}{s}}}} = 0} & (138) \end{matrix}$

This is nothing but the method of weighted residual. Generally, a variation operation using this equation in which δ{tilde over ( )}u_(i) and δu_(i) are regarded as identical is performed, and a mechanical boundary condition and a differential equation are obtained. However, this is benefit by eq. (123).

(3) Once the benefit of eq. (123) is considered effects by the principle of virtual work and is placed at the basis of the system of dynamics, nothing but a self-adjoint problem can be dealt with. Even if a problem is a non-self-adjoint problem, the problem can be solved by a function of the method of weighted residual in some cases. Therefore, this issue is hardly noticeable. Ambiguous parts of the principle of virtual work should be reconsidered on the basis of the present chapter (1), particularly eq. (122). With this, various types of dynamics such as the theory of elasticity can be expected to be reconstructed into a better system and achieve new development.

(4) Generally, a linear elastic finite element is formulated by eq. (132), but with respect to an h-element, an operation of a variation is not performed. When the variation sign δ is removed from eq. (132) and the equation is multiplied by ½, the obtained equation becomes in accordance with the energy conservation law of eq. (134), and the h-element is dealt with according to this. If an operation of a variation by eq. (123) is performed, an element equivalent to that of the p-method is obtained, and a new finite element can be configured by eqs. (122) and (127).

8. New Finite Element

8.1 Shape function

For example, let node numbers of a 16-nodes quadrilateral element be expressed as shown in FIG. 8. Then, we obtain the following shape functions N_(i) (i=1 to 16) thereof:

[Formula 81]

N ₁≡− 1/12(ξ−1)(η−1)(4ξ³−ξ+4η³−η+3)  (139)

N ₂≡− 1/12(ξ+1)(η−1)(4ξ³−ξ−4η³+η−3)  (140)

N ₃≡+ 1/12(ξ+1)(η+1)(4ξ³−ξ+4η³−η−3)  (141)

N ₄≡+ 1/12(ξ−1)(η+1)(4ξ³−ξ−4η³+η+3)  (142)

N ₅≡+⅔(ξ²−1)ξ(2ξ−1)(η−1)  (143)

N ₆≡−½(ξ²−1)(4ξ²−1)(η−1)  (144)

N ₇≡+⅔(ξ²−1)ξ(2ξ−1)(η−1)  (145)

N ₈≡−⅔(ξ+1)(η²−1)η(2η−1)  (146)

N ₉≡+½(ξ+1)(η²−1)η(4η²−1)  (147)

N ₁₀≡−⅔(ξ+1)(η²−1)η(2η+1)  (148)

N ₁₁≡−⅔(ξ²−1)ξ(2ξ+1)(η+1)  (149)

N ₁₂≡+½(ξ²−1)(4ξ²−1)(η+1)  (150)

N ₁₃≡−⅔(ξ²−1)ξ(2ξ−1)(η+1)  (151)

N ₁₄≡+⅔(ξ−1)(η²−1)η(2η+1)  (152)

N ₁₅≡−½(ξ−1)(η²−1)(4η²−1)  (153)

N ₁₆≡+⅔(ξ−1)(η²−1)η(2η−1)  (154)

These equations represent insides of boundaries when ξ and η are in the ranges indicated by the signs of inequality of the following expressions, and represent boundaries when ξ and η are in the ranges indicated by the equal signs of the following expressions:

−1≦ξ≦1

−1≦η≦1)

8.2 Coordinates Inside Element

With use of a shape function N_(i) (i=1 to n{tilde over ( )}), a coordinate x_(j) in the inside of an element is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 82} \right\rbrack & \; \\ {x_{j} \equiv {\sum\limits_{i = 1}^{\overset{\sim}{n}}{N_{i}X_{ij}}}} & (156) \end{matrix}$

Here, X_(ij) represents a coordinate of the node and fir, represents the number of nodes of the element. The details of coordinates x, y is represented as following.

$\begin{matrix} {\begin{Bmatrix} x \\ y \end{Bmatrix} \equiv \begin{Bmatrix} x_{1} \\ x_{2} \end{Bmatrix} \equiv {\begin{bmatrix} N_{1} & 0 & \ldots & N_{\overset{\sim}{n}} & 0 \\ 0 & N_{1} & \ldots & 0 & N_{\overset{\sim}{n}} \end{bmatrix}\begin{Bmatrix} X_{11} \\ X_{12} \\ \vdots \\ X_{\overset{\sim}{n}\; 1} \\ X_{\overset{\sim}{n}\; 2} \end{Bmatrix}}} & (157) \end{matrix}$

Given

n≡2 n  (158)

the foregoing equation is expressed with matrices as:

$\begin{matrix} {\underset{2 \times 1}{\left\{ x \right\}} \equiv {\underset{2 \times n}{\lbrack N\rbrack}\underset{n \times 1}{\left\{ X \right\}}}} & (159) \end{matrix}$

8.3 Internal Displacement of Element

The displacement, which is to be a primal solution, is divided into a term representing a boundary displacement and a term such that the displacement becomes zero at boundary, and the terms are expressed as

[Formula 83]

u _(x) ≡u ₁ ≡u _(A1) +u _(o1)  (160)

u _(y) ≡u ₂ ≡u _(A2) +u _(o2)  (161)

The index A indicates that the term represents a boundary displacement, and the index o indicates that the term is such that the displacement becomes zero at boundary. They are combined as a displacement function u_(j) into:

[Formula 84]

u _(j) ≡u _(Aj) +u _(oj)  (162)

This is expressed with matrices as follows:

$\begin{matrix} {\underset{2 \times 1}{\left\{ u \right\}} \equiv {\underset{2 \times 1}{\left\{ u_{A} \right\}} + \underset{2 \times 1}{\left\{ u_{o} \right\}}}} & (163) \end{matrix}$

With use of an interpolation function χ_(Ai) (i=1 to n{tilde over ( )}), an internal displacement u_(Aj) of the element is written as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 85} \right\rbrack & \; \\ {u_{Aj} \equiv {\sum\limits_{i = 1}^{\overset{\sim}{n}}{\chi_{Ai}U_{ij}}}} & (164) \end{matrix}$

U_(ij) represents a nodal displacement. Here, as is the case with an isoparametric element, the interpolation function (interpolation function) χ_(Ai) is caused to coincide with a shape function N_(i). The details of the displacement are as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 86} \right\rbrack & \; \\ {\begin{Bmatrix} u_{A\; 1} \\ u_{A\; 2} \end{Bmatrix} \equiv {\begin{bmatrix} _{A\; 1} & 0 & \ldots & _{A\overset{\sim}{n}} & 0 \\ 0 & _{A\; 1} & \ldots & 0 & _{A\overset{\sim}{n}} \end{bmatrix}\begin{Bmatrix} U_{11} \\ U_{12} \\ \vdots \\ U_{\overset{\sim}{n\;}1} \\ U_{\overset{\sim}{n}\; 2} \end{Bmatrix}}} & (165) \end{matrix}$

This is expressed with matrices as follows:

$\begin{matrix} {\underset{2 \times 1}{\left\{ u_{A} \right\}} \equiv {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}} & (166) \end{matrix}$

u_(Aj) represents not only a displacement on the element boundary but also an internal displacement of the element, but there is no guarantee that this satisfies the differential equation. Since the ability of u_(Aj) for representing the internal displacement distribution of an element is limited, only the ability of representing a displacement at an element boundary is expected to be reliable. In this sense, u_(Aj) is referred to as a “boundary function”. It is expected that an internal displacement of an element that is beyond representation by u_(Aj) is corrected by u_(oj). In this sense, u_(oj) is referred to as a “correction function”. That is, eq. (162) represents the displacement function u_(j) by sum of the boundary function u_(Aj) and correction function u_(oj).

The correction function u_(oj) is expressed with a sum of a trial function ψ_(ok) (k=1 to l{tilde over ( )}) such that displacement at boundary is zero, which is written as:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 87} \right\rbrack} & \; \\ {\mspace{79mu} {u_{o\; 1} \equiv {\text{?}c_{{ok}\; 1}\psi_{ok}}}} & (167) \\ {\mspace{79mu} {{u_{o\; 2} \equiv {\text{?}c_{{ok}\; 2}\psi_{ok}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (168) \end{matrix}$

They are combined into:

$\begin{matrix} {\mspace{79mu} {{u_{oj} \equiv {\text{?}c_{okj}\psi_{ok}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (169) \end{matrix}$

The details of the correction function are as follows:

$\begin{matrix} {\mspace{79mu} {{\begin{Bmatrix} u_{o\; 1} \\ u_{o\; 2} \end{Bmatrix} \equiv {\begin{bmatrix} \psi_{o\; 1} & 0 & \ldots & \text{?} & 0 \\ 0 & \psi_{o\; 1} & \ldots & 0 & \text{?} \end{bmatrix}\begin{Bmatrix} c_{o\; 11} \\ c_{o\; 12} \\ \vdots \\ \text{?} \\ \text{?} \end{Bmatrix}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (170) \end{matrix}$

When the following is given:

[Formula 88]

λ≡2,  (171)

then, eq. (170) is expressed with matrices as:

$\begin{matrix} {\underset{2 \times 1}{\left\{ u_{o} \right\}} \equiv {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} & (172) \end{matrix}$

Consequently, we obtain:

$\begin{matrix} {\underset{2 \times 1}{\left\{ u \right\}} \equiv {\underset{2 \times 1}{\left\{ u_{A} \right\}} + \underset{2 \times 1}{\left\{ u_{o} \right\}}} \equiv {{\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}}} & (173) \end{matrix}$

On the other hand, in eq. (24), a sum of the term U_(Bj) satisfying the inhomogeneous boundary condition and the term u_(Hj) satisfying the homogeneous boundary condition, represents a displacement u_(j), which is to be a primal solution. This is expressed with matrices as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 89} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u \right\}} \equiv {\underset{2 \times 1}{\left\{ u_{B} \right\}} + \underset{2 \times 1}{\left\{ u_{H} \right\}}}} & (174) \end{matrix}$

At the stage where a boundary condition is given, details thereof are determined.

8.4 Trial Function

As the trial function ψ_(ok) (k=1 to l{tilde over ( )}), a function system such that a displacement at boundary is zero is used. For example, it may be the associated Legendre function (associated Legendre function), or the trigonometric function. In the case of the trigonometric function, let the function system be:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 90} \right\rbrack & \; \\ {{f_{si}(\xi)} \equiv {\sin \left\{ {\left( {2i} \right)\frac{\pi}{2}\xi} \right\}}} & (175) \\ {{f_{ci}(\xi)} \equiv {\cos \left\{ {\left( {{2i} - 1} \right)\frac{\pi}{2}\xi} \right\}}} & (176) \\ {{g_{si}(\eta)} \equiv {\sin \left\{ {\left( {2i} \right)\frac{\pi}{2}\eta} \right\}}} & (177) \\ {{g_{ci}(\eta)} \equiv {\cos \left\{ {\left( {{2i} - 1} \right)\frac{\pi}{2}\eta} \right\}}} & (178) \end{matrix}$

Then, a product of the same as follows is used as the trial function ψ_(ok):

$\begin{matrix} {\begin{pmatrix} {f_{si}(\xi)} \\ {f_{ci}(\xi)} \end{pmatrix} \times \begin{pmatrix} {g_{sj}(\xi)} \\ {g_{cj}(\eta)} \end{pmatrix}} & (179) \end{matrix}$

8.5 Dual Displacement

The dual displacement u_(j*), which is to be a dual solution, is divided into a term representing a boundary displacement, and a term such that the displacement is zero at boundary, and is written as:

[Formula 91]

u _(j) *≡u _(Aj) *+u _(oj)*  (180)

This is expressed with matrices as follows:

$\begin{matrix} {\underset{2 \times 1}{\left\{ u^{*} \right\}} \equiv {\underset{2 \times 1}{\left\{ u_{A}^{*} \right\}} + \underset{2 \times 1}{\left\{ u_{o}^{*} \right\}}}} & (181) \end{matrix}$

With use of an interpolation function χ_(Ai), an internal displacement u*_(Aj) in an element is written as:

$\begin{matrix} {u_{Aj}^{*} \equiv {\sum\limits_{i = 1}^{\overset{\sim}{n}}\; {_{Ai}U_{ij}^{*}}}} & (182) \end{matrix}$

U_(ij)* represents a nodal displacement. Here, like the method currently used for isoparametric element, the shape function N_(i) is adopted as the interpolation function χ_(Ai). The details of the displacement are as follows:

$\begin{matrix} {\begin{Bmatrix} u_{A\; 1}^{*} \\ u_{A\; 2}^{*} \end{Bmatrix} \equiv {\begin{bmatrix} _{A\; 1} & 0 & \ldots & _{A\overset{\sim}{n}} & 0 \\ 0 & _{A\; 1} & \ldots & 0 & _{A\overset{\sim}{n}} \end{bmatrix}\begin{Bmatrix} U_{11}^{*} \\ U_{12}^{*} \\ \vdots \\ U_{\overset{\sim}{n}\; 1}^{*} \\ U_{\overset{\sim}{n\;}2}^{*} \end{Bmatrix}}} & (183) \end{matrix}$

This is expressed with matrices as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 92} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u_{A}^{*} \right\}} \equiv {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}}} & (184) \end{matrix}$

The correction function u_(oj*) is expressed with a sum of a trial function ψ_(ok) (k=1 to l{tilde over ( )}), which is written as:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 93} \right\rbrack} & \; \\ {\mspace{79mu} {u_{o\; 1}^{*} \equiv {\text{?}c_{{ok}\; 1}^{*}\psi_{ok}}}} & (185) \\ {\mspace{79mu} {{u_{o\; 2}^{*} \equiv {\text{?}c_{{ok}\; 2}^{*}\psi_{ok}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (186) \end{matrix}$

They are combined into:

$\begin{matrix} {u_{oj}^{*} \equiv {\sum\limits_{k = 1}^{\overset{\sim}{}}{c_{okj}^{*}\psi_{ok}}}} & (187) \end{matrix}$

The details of the correction function are as follows:

$\begin{matrix} {\begin{Bmatrix} u_{o\; 1}^{*} \\ u_{o\; 2}^{*} \end{Bmatrix} \equiv {\begin{bmatrix} \psi_{o\; 1} & 0 & \Lambda & \psi_{o\overset{\sim}{}} & 0 \\ 0 & \psi_{o\; 1} & \Lambda & 0 & \psi_{o\overset{\sim}{}} \end{bmatrix}\begin{Bmatrix} c_{o\; 11}^{*} \\ c_{o\; 12}^{*} \\ M \\ c_{o\; \overset{\sim}{}1}^{*} \\ c_{o\; \overset{\sim}{}2}^{*} \end{Bmatrix}}} & (188) \end{matrix}$

This equation is expressed with matrices as follows:

$\begin{matrix} {\underset{2 \times 1}{\left\{ u_{o}^{*} \right\}} \equiv {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} & (189) \end{matrix}$

Consequently, we obtain:

$\begin{matrix} {\underset{2 \times 1}{\left\{ u^{*} \right\}} \equiv {\underset{2 \times 1}{\left\{ u_{A}^{*} \right\}} + \underset{2 \times 1}{\left\{ u_{o}^{*} \right\}}} \equiv {{\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}}} & (190) \end{matrix}$

On the other hand, in eq. (42), a sum of the term u_(Bj*) satisfying the inhomogeneous adjoint boundary condition and the term u_(Hj*) satisfying the homogeneous adjoint boundary condition, represents a dual displacement u_(j*). This is expressed with matrices as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 94} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u^{*} \right\}} \equiv {\underset{2 \times 1}{\left\{ u_{B}^{*} \right\}} + \underset{2 \times 1}{\left\{ u_{H}^{*} \right\}}}} & (191) \end{matrix}$

At the stage where a boundary condition is given, details thereof are determined.

8.6 Equivalent Nodal Force

From eqs. (4) to (6), a strain component is given as:

$\begin{matrix} {\begin{Bmatrix} ɛ_{x} \\ ɛ_{y} \\ \gamma_{xy} \end{Bmatrix} \equiv {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{bmatrix}\begin{Bmatrix} {\frac{\partial}{\partial x}u_{x}} \\ {\frac{\partial}{\partial x}u_{y}} \\ {\frac{\partial}{\partial y}u_{x}} \\ {\frac{\partial}{\partial y}u_{y}} \end{Bmatrix}}} & (192) \end{matrix}$

This is expressed with matrices as follows:

$\begin{matrix} {\underset{3 \times 1}{\left\{ ɛ \right\}} \equiv {\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du} \right\}}}} & (193) \end{matrix}$

From eqs. (7) to (9), a stress component is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 95} \right\rbrack & \; \\ {\begin{Bmatrix} \sigma_{x} \\ \sigma_{y} \\ \tau_{xy} \end{Bmatrix} = {{G\begin{bmatrix} {\mu + 1} & {\mu - 1} & 0 \\ {\mu - 1} & {\mu + 1} & 0 \\ 0 & 0 & 1 \end{bmatrix}}\begin{Bmatrix} ɛ_{x} \\ ɛ_{y} \\ \gamma_{xy} \end{Bmatrix}}} & (194) \end{matrix}$

This is expressed with matrices as follows:

$\begin{matrix} {\underset{3 \times 1}{\left\{ \sigma \right\}} = {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 1}{\left\{ ɛ \right\}}}} & (195) \end{matrix}$

Consequently, we obtain:

$\begin{matrix} {\underset{3 \times 1}{\left\{ \sigma \right\}} = {{G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 1}{\left\{ ɛ \right\}}} \equiv {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du} \right\}}}}} & (196) \end{matrix}$

This represents the stress component of eqs. (10) to (12). Calculating a coefficient matrix, we obtain:

$\begin{matrix} {{\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}} = {{\begin{bmatrix} {\mu + 1} & {\mu - 1} & 0 \\ {\mu - 1} & {\mu + 1} & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{bmatrix}} = \begin{bmatrix} {\mu + 1} & 0 & 0 & {\mu - 1} \\ {\mu - 1} & 0 & 0 & {\mu + 1} \\ 0 & 1 & 1 & 0 \end{bmatrix}}} & (197) \end{matrix}$

A differential coefficient of the displacement is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 96} \right\rbrack & \; \\ \begin{matrix} {\underset{4 \times 1}{\left\{ {Du} \right\}} \equiv \begin{Bmatrix} {\frac{\partial}{\partial x}u_{x}} \\ {\frac{\partial}{\partial x}u_{y}} \\ {\frac{\partial}{\partial y}u_{x}} \\ {\frac{\partial}{\partial x}u_{y}} \end{Bmatrix}} \\ {\equiv \begin{Bmatrix} {\underset{2 \times 1}{\frac{\partial}{\partial x}\left\{ u_{A} \right\}} + \underset{2 \times 1}{\frac{\partial}{\partial x}\left\{ u_{o} \right\}}} \\ {\underset{2 \times 1}{\frac{\partial}{\partial y}\left\{ u_{A} \right\}} + \underset{2 \times 1}{\frac{\partial}{\partial y}\left\{ u_{o} \right\}}} \end{Bmatrix}} \\ {\equiv {{\begin{bmatrix} \underset{2 \times n}{\frac{\partial}{\partial x}\left\lbrack \chi_{A} \right\rbrack} \\ \underset{2 \times n}{\frac{\partial}{\partial y}\left\lbrack \chi_{A} \right\rbrack} \end{bmatrix}\underset{n \times 1}{\left\{ U \right\}}} + {\begin{bmatrix} \underset{2 \times \lambda}{\frac{\partial}{\partial x}\left\lbrack \psi_{o} \right\rbrack} \\ \underset{2 \times \lambda}{\frac{\partial}{\partial y}\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}}} \end{matrix} & (198) \end{matrix}$

Let the matrices on the right side of the equation be given as:

$\begin{matrix} {\mspace{20mu} \left\lbrack {{Formula}\mspace{14mu} 97} \right\rbrack} & \; \\ {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack} \equiv \begin{bmatrix} \underset{2 \times n}{\frac{\partial}{\partial x}\left\lbrack \chi_{A} \right\rbrack} \\ \underset{2 \times n}{\frac{\partial}{\partial y}\left\lbrack \chi_{A} \right\rbrack} \end{bmatrix} \equiv \begin{bmatrix} {\frac{\partial}{\partial x}\chi_{A\; 1}} & 0 & \Lambda & {\frac{\partial}{\partial x}\chi_{A\; n\overset{\sim}{}}} & 0 \\ 0 & {\frac{\partial}{\partial x}\chi_{A\; 1}} & \Lambda & 0 & {\frac{\partial}{\partial x}\chi_{{An}\; \overset{\sim}{}}} \\ {\frac{\partial}{\partial y}\chi_{A\; 1}} & 0 & \Lambda & {\frac{\partial}{\partial y}\chi_{A\; n\overset{\sim}{}}} & 0 \\ 0 & {\frac{\partial}{\partial y}\chi_{A\; 1}} & \Lambda & 0 & {\frac{\partial}{\partial y}\chi_{{An}\overset{\sim}{}}} \end{bmatrix}}\mspace{20mu} {and}} & (199) \\ {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack} \equiv \begin{bmatrix} \underset{2 \times \lambda}{\frac{\partial}{\partial x}\left\lbrack \psi_{o} \right\rbrack} \\ \underset{2 \times \lambda}{\frac{\partial}{\partial y}\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix} \equiv \begin{bmatrix} {\frac{\partial}{\partial x}\psi_{o\; 1}} & 0 & \Lambda & {\frac{\partial}{\partial x}\psi_{o\overset{\sim}{}}} & 0 \\ 0 & {\frac{\partial}{\partial x}\psi_{o\; 1}} & \Lambda & 0 & {\frac{\partial}{\partial x}\psi_{o\overset{\sim}{}}} \\ {\frac{\partial}{\partial y}\psi_{{o\; 1}\;}} & 0 & \Lambda & {\frac{\partial}{\partial y}\psi_{o\overset{\sim}{}}} & 0 \\ 0 & {\frac{\partial}{\partial y}\psi_{o\; 1}} & \Lambda & 0 & {\frac{\partial}{\partial y}\psi_{o\overset{\sim}{}}} \end{bmatrix}} & (200) \end{matrix}$

Then, the differential coefficient of the displacement is expressed as:

$\begin{matrix} {\underset{4 \times 1}{\left\{ {Du} \right\}} \equiv {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}}} & (201) \end{matrix}$

The stress is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 98} \right\rbrack & \; \\ {\underset{3 \times 1}{\left\{ \sigma \right\}} \equiv {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du} \right\}}} \equiv {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\left( {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} \right)}} & (202) \end{matrix}$

The non-dimensional stress is given as:

$\begin{matrix} {\underset{3 \times 1}{\left\{ {\sigma/G} \right\}} \equiv {\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du} \right\}}} \equiv {\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\left( {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} \right)}} & (203) \end{matrix}$

The relationship between a surface force vector and a stress tensor is given as:

$\begin{matrix} {\begin{Bmatrix} p_{x} \\ p_{y} \end{Bmatrix} \equiv {\begin{bmatrix} n_{x} & 0 & n_{y} \\ 0 & n_{y} & n_{x} \end{bmatrix}\begin{Bmatrix} \sigma_{x} \\ \sigma_{y} \\ \tau_{xy} \end{Bmatrix}}} & (204) \end{matrix}$

Let the following be given as:

$\begin{matrix} {\underset{2 \times 3}{\lbrack T\rbrack} \equiv \underset{2 \times 3}{\begin{bmatrix} n_{x} & 0 & n_{y} \\ 0 & n_{y} & n_{x} \end{bmatrix}}} & (205) \end{matrix}$

Then, the relationship is expressed as:

$\begin{matrix} {\underset{2 \times 1}{\left\{ p \right\}} \equiv {\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 1}{\left\{ \sigma \right\}}}} & (206) \end{matrix}$

Therefore, the surface force vector is given as:

$\begin{matrix} {\underset{2 \times 1}{\left\{ p \right\}} \equiv {\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 1}{\left\{ \sigma \right\}}} \equiv {G\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\left( {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} \right)}} & (207) \end{matrix}$ [Formula 99]

A displacement on a boundary surface of an element is:

$\begin{matrix} {\underset{{2 \times 1}\;}{\left\{ u_{A} \right\}} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}} & (208) \end{matrix}$

The work wrk done by the surface force vector is:

$\begin{matrix} {{wrk} \equiv {\frac{1}{2}{\underset{1 \times 2}{\left\{ U_{A} \right\}}}^{T}\underset{2 \times 1}{\left\{ p \right\}}}} & (209) \end{matrix}$

Therefore, we obtain:

$\begin{matrix} {{wrk} \equiv {\frac{1}{2}G{\underset{1 \times n}{\left\{ U \right\}}}^{T}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T}\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\left( {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} \right)}} & (210) \end{matrix}$

The integration on the boundary of the element gives work per unit thickness as follows:

$\begin{matrix} {{\int_{C}^{\;}{{wrk}\ {c}\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\frac{1}{2}G{\underset{1 \times n}{\left\{ U \right\}}}^{T}\left( {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} \right)}}{where}} & (211) \\ {\underset{n \times n}{\left\lbrack K_{U} \right\rbrack} \equiv {\int_{C}^{\;}{{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T}\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\ {c}}}} & (212) \\ {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack} \equiv {\int_{C}^{\;}{{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T}\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\ {c}}}} & (213) \end{matrix}$

Recognizing the inside in the parentheses on the right side of the equation (211) as an equivalent nodal force F per unit thickness, we obtain:

$\begin{matrix} {\underset{n \times 1}{\left\{ F \right\}} \equiv {G\left( {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} \right)}} & (214) \end{matrix}$

In the case where a plane stress state is dealt with, this equation is multiplied by a plate thickness h, and we obtain an equivalent nodal force given as:

$\begin{matrix} {\underset{n \times 1}{\left\{ {Fh} \right\}} \equiv {{Gh}\left( {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} \right)}} & (215) \end{matrix}$

Further, it is useful to recognize:

$\begin{matrix} {\underset{n \times 1}{\left\{ {F/G} \right\}} \equiv {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}}} & (216) \end{matrix}$

8.7 Dual Equivalent Nodal Force

From eqs. (4) to (6), a strain component is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 100} \right\rbrack & \; \\ {\begin{Bmatrix} ɛ_{x}^{*} \\ ɛ_{y}^{*} \\ \gamma_{xy}^{*} \end{Bmatrix} \equiv {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{bmatrix}\begin{Bmatrix} {\frac{\partial\;}{\partial x}u_{x}^{*}} \\ {\frac{\partial\;}{\partial x}u_{y}^{*}} \\ {\frac{\partial\;}{\partial y}u_{x}^{*}} \\ {\frac{\partial\;}{\partial y}u_{y}^{*}} \end{Bmatrix}}} & (217) \end{matrix}$

This is expressed with matrices as follows:

$\begin{matrix} {\underset{3 \times 1}{\left\{ ɛ^{*} \right\}} \equiv {\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du}^{*} \right\}}}} & (218) \end{matrix}$

From eqs. (7) to (9), the stress component is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 101} \right\rbrack & \; \\ {\begin{Bmatrix} \sigma_{x}^{*} \\ \sigma_{y}^{*} \\ \tau_{xy}^{*} \end{Bmatrix} = {{G\begin{bmatrix} {\mu + 1} & {\mu - 1} & 0 \\ {\mu - 1} & {\mu + 1} & 0 \\ 0 & 0 & 1 \end{bmatrix}}\begin{Bmatrix} ɛ_{x}^{*} \\ ɛ_{y}^{*} \\ \gamma_{xy}^{*} \end{Bmatrix}}} & (219) \end{matrix}$

This is expressed with matrices as follows:

$\begin{matrix} {\underset{3 \times 1}{\left\{ \sigma^{*} \right\}} = {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 1}{\left\{ ɛ^{*} \right\}}}} & (220) \end{matrix}$

Therefore, we obtain:

$\begin{matrix} {\underset{3 \times 1}{\left\{ \sigma^{*} \right\}} = {{G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 1}{\left\{ ɛ^{*} \right\}}} \equiv {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du}^{*} \right\}}}}} & (221) \end{matrix}$

This represents the stress components of eqs. (10) to (12).

The differential coefficient of the displacement is given as:

$\begin{matrix} \begin{matrix} {\underset{4 \times 1}{\left\{ {Du}^{*} \right\}} \equiv \begin{Bmatrix} {\frac{\partial}{\partial x}u_{x}^{*}} \\ {\frac{\partial}{\partial x}u_{y}^{*}} \\ {\frac{\partial}{\partial y}u_{x}^{*}} \\ {\frac{\partial}{\partial y}u_{y}^{*}} \end{Bmatrix}} \\ {\equiv \begin{Bmatrix} {\underset{2 \times 1}{\frac{\partial}{\partial x}\left\{ u_{A}^{*} \right\}} + \underset{2 \times 1}{\frac{\partial}{\partial x}\left\{ u_{0}^{*} \right\}}} \\ {\underset{2 \times 1}{\frac{\partial}{\partial y}\left\{ u_{A}^{*} \right\}} + \underset{2 \times 1}{\frac{\partial}{\partial y}\left\{ u_{o}^{*} \right\}}} \end{Bmatrix}} \\ {\equiv {{\begin{bmatrix} \underset{2 \times n}{\frac{\partial}{\partial x}\left\lbrack \chi_{A} \right\rbrack} \\ \underset{2 \times n}{\frac{\partial}{\partial y}\left\lbrack \chi_{A} \right\rbrack} \end{bmatrix}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\begin{bmatrix} \underset{2 \times \lambda}{\frac{\partial}{\partial x}\left\lbrack \psi_{o} \right\rbrack} \\ \underset{2 \times \lambda}{\frac{\partial}{\partial y}\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}}} \end{matrix} & (222) \end{matrix}$

Using eqs. (199) and (200), we obtain:

$\begin{matrix} {\underset{4 \times 1}{\left\{ {Du}^{*} \right\}} \equiv {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}}} & (223) \end{matrix}$

The stress is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 103} \right\rbrack & \; \\ {\underset{3 \times 1}{\left\{ \sigma^{*} \right\}} \equiv {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du}^{*} \right\}}} \equiv {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\left( {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} \right)}} & (224) \end{matrix}$

The non-dimensional stress is given as:

$\begin{matrix} {\underset{3 \times 1}{\left\{ {\sigma^{*}/G} \right\}} \equiv {\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du}^{*} \right\}}} \equiv {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\left( {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} \right)}} & (225) \end{matrix}$

The relationship between the surface force vector and the stress tensor is given as:

$\begin{matrix} {\begin{Bmatrix} p_{x}^{*} \\ p_{y}^{*} \end{Bmatrix} \equiv {\begin{bmatrix} n_{x} & 0 & n_{y} \\ 0 & n_{y} & n_{x} \end{bmatrix}\begin{Bmatrix} \sigma_{x}^{*} \\ \sigma_{y}^{*} \\ \tau_{xy}^{*} \end{Bmatrix}}} & (226) \end{matrix}$

This is expressed as:

$\begin{matrix} {\underset{2 \times 1}{\left\{ p^{*} \right\}} \equiv {\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 1}{\left\{ \sigma^{*} \right\}}}} & (227) \end{matrix}$

Therefore, the surface force vector is given as:

$\begin{matrix} {\underset{2 \times 1}{\left\{ p^{*} \right\}} \equiv {\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 1}{\left\{ \sigma^{*} \right\}}} \equiv {G\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\left( {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} \right)}} & (228) \end{matrix}$

A displacement in an element boundary surface is:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 104} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u_{A}^{*} \right\}} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}}} & (229) \end{matrix}$

The work wrk* done by the surface force vector is:

$\begin{matrix} {{wrk}^{*} \equiv {\frac{1}{2}\underset{1 \times 2}{\left\{ U_{A}^{*} \right\}^{T}}\underset{2 \times 1}{\left\{ p^{*} \right\}}}} & (230) \end{matrix}$

Therefore, we obtain:

$\begin{matrix} {{wrk}^{*} \equiv {\frac{1}{2}G\underset{1 \times n}{\left\{ U^{*} \right\}^{T}}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T}\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\left( {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} \right)}} & (231) \end{matrix}$

The integration on the boundary of the element and use of eqs. (212) and (213) gives the work per unit thickness as follows:

$\begin{matrix} {{\int_{C}{{wrk}^{*}{c}}} \equiv {\frac{1}{2}G\underset{1 \times n}{\left\{ U^{*} \right\}^{T}}\left( {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} \right)}} & (232) \end{matrix}$

Recognizing the inside in the parentheses on the right side of the equation (232) as a dual equivalent nodal force F* per unit thickness, we obtain:

$\begin{matrix} {\underset{n \times 1}{\left\{ F^{*} \right\}} \equiv {G\left( {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} \right)}} & (233) \end{matrix}$

In the case where a plane stress state is dealt with, this equation is multiplied by a plate thickness h, and we obtain a dual equivalent nodal force as follows:

$\begin{matrix} {\underset{n \times 1}{\left\{ {F^{*}/G} \right\}} \equiv {{Gh}\left( {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} \right)}} & (234) \end{matrix}$

Further, it is also useful to recognize it as follows:

$\begin{matrix} {\underset{n \times 1}{\left\{ {F^{*}/G} \right\}} \equiv {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}}} & (235) \end{matrix}$

8.8 Integration on Boundary

A specific procedure for executing integration on each side is as follows.

[Formula 105]

The differential of the length on each side is as follows:

$\begin{matrix} {{{c} \equiv \sqrt{\left( {x} \right)^{2} + \left( {y} \right)^{2}}} = \sqrt{\left( {{\frac{\partial x}{\partial\xi}{\xi}} + {\frac{\partial x}{\partial\eta}{\eta}}} \right)^{2} + \left( {{\frac{\partial y}{\partial\xi}{\xi}} + {\frac{\partial y}{\partial\eta}{\eta}}} \right)^{2}}} & (236) \end{matrix}$

(1) Giving dη=0 on a curve of η=Const., we obtain:

$\begin{matrix} {{{c} \equiv {J_{\xi}{\xi}}};{J_{\xi} \equiv \sqrt{\left( \frac{\partial x}{\partial\xi} \right)^{2} + \left( \frac{\partial y}{\partial\xi} \right)^{2}}}} & (237) \end{matrix}$

Integration on the curve gives:

$\begin{matrix} {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack} \equiv {\int_{- 1}^{1}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack^{T}}\underset{2 \times 3}{\lbrack T\rbrack}\ \underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}J_{\xi}{\xi}}}}{\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack} \equiv {\int_{- 1}^{1}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack^{T}}\underset{2 \times 3}{\lbrack T\rbrack}\ \underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}J_{\xi}{\xi}}}}} & (238) \end{matrix}$

(2) Giving dξ=0 on a curve of ξ=Const., we obtain:

$\begin{matrix} {{{c} \equiv {J_{\eta}{\eta}}};{J_{\eta} \equiv \sqrt{\left( \frac{\partial x}{\partial\eta} \right)^{2} + \left( \frac{\partial y}{\partial\eta} \right)^{2}}}} & (239) \end{matrix}$

Integration on the curve gives:

$\begin{matrix} {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack} \equiv {\int_{- 1}^{1}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack^{T}}\underset{2 \times 3}{\lbrack T\rbrack}\ \underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}J_{\eta}{\eta}}}}{\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack} \equiv {\int_{- 1}^{1}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack^{T}}\underset{2 \times 3}{\lbrack T\rbrack}\ \underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}J_{\eta}{\eta}}}}} & (240) \end{matrix}$

8.9 Strain Energy

Internal strain energy eng of an element is given as:

$\begin{matrix} {{{eng} \equiv {\frac{1}{2}\underset{1 \times 3}{\left\{ ɛ \right\}^{T}}\underset{3 \times 1}{\left\{ \sigma \right\}}}},} & (241) \end{matrix}$

which is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 106} \right\rbrack & \; \\ {{eng} \equiv {\frac{1}{2}G\underset{1 \times 4}{\left\{ {Du} \right\}^{T}}{\underset{4 \times 3}{\lbrack d\rbrack}}^{T}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du} \right\}}}} & (242) \end{matrix}$

Therefore, we obtain:

$\begin{matrix} {{eng} \equiv {\frac{1}{2}{G\left( {{\underset{1 \times n}{\left\{ U \right\}^{T}}{\underset{n \times 4}{\left\lbrack {D\; \chi_{A}} \right\rbrack}}^{T}} + {\underset{1 \times \eta}{\left\{ c_{o} \right\}^{T}}{\underset{\lambda \times 4}{\left\lbrack {D\; \psi_{o}} \right\rbrack}}^{T}}} \right)}{\underset{4 \times 3}{\lbrack d\rbrack}}^{T}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\left( {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} \right)}} & (243) \end{matrix}$

8.10 Dual Strain Energy

Internal strain energy eng* in an element is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 107} \right\rbrack & \; \\ {{eng}^{*} \equiv {\frac{1}{2}\underset{1 \times 3}{\left\{ ɛ^{*} \right\}^{T}}\underset{3 \times 1}{\left\{ \sigma^{*} \right\}}}} & (244) \end{matrix}$

which is given as:

$\begin{matrix} {{eng}^{*} \equiv {\frac{1}{2}G\underset{1 \times 4}{\left\{ {Du}^{*} \right\}^{T}}{\underset{4 \times 3}{\lbrack d\rbrack}}^{T}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du}^{*} \right\}}}} & (245) \end{matrix}$

Therefore, we obtain:

$\begin{matrix} {{eng}^{*} \equiv {\frac{1}{2}{G\left( {{\underset{1 \times n}{\left\{ U^{*} \right\}^{T}}{\underset{n \times 4}{\left\lbrack {D\; \chi_{A}} \right\rbrack}}^{T}} + {\underset{1 \times \eta}{\left\{ c_{o}^{*} \right\}^{T}}{\underset{\lambda \times 4}{\left\lbrack {D\; \psi_{o}} \right\rbrack}}^{T}}} \right)}{\underset{4 \times 3}{\lbrack d\rbrack}}^{T}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\left( {{\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} \right)}} & (246) \end{matrix}$

8.11 Adjoint Boundary Condition

The displacement on a boundary surface is expressed by eq. (208), the dual displacement is expressed by eq. (229), the surface force is expressed by eq. (207), and the dual surface force is expressed by eq. (228). With use of these, the boundary term R of eq. (60) is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 108} \right\rbrack & \; \\ \begin{matrix} {R \equiv {\frac{1}{G}{\sum\limits_{i}\; {\int_{C}{\left( {{u_{i}^{*}p_{i}} - {u_{i}p_{i}^{*}}} \right)\ {c}}}}}} \\ {= {\frac{1}{G}{\int_{C}{\left( {{\underset{1 \times 2}{\left\{ u_{A}^{*} \right\}^{T}}\underset{2 \times 1}{\left\{ p \right\}}} - {\underset{1 \times 2}{\left\{ u_{A} \right\}^{T}}\underset{2 \times 1}{\left\{ p^{*} \right\}}}}\  \right){c}}}}} \end{matrix} & (247) \end{matrix}$

The integration on the boundary of the element and the use of eqs. (212) and (213) gives:

$\begin{matrix} {R \equiv {{\underset{1 \times n}{\left\{ U^{*} \right\}^{T}}\left( {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} \right)} - {\underset{1 \times n}{\left\{ U \right\}^{T}}\left( {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} \right)}}} & (248) \end{matrix}$

With use of eqs. (216) and (235) expressing the equivalent nodal force, this equation is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 109} \right\rbrack & \; \\ {R \equiv {{\underset{1 \times n}{\left\{ U^{*} \right\}^{T}}\underset{n \times 1}{\left\{ {F/G} \right\}}} - {\underset{1 \times n}{\left\{ U \right\}^{T}}\underset{n \times 1}{\left\{ {F^{*}/G} \right\}}}}} & (249) \end{matrix}$

This is transformed into:

$\begin{matrix} {R \equiv {\begin{Bmatrix} \underset{1 \times n}{\left\{ {F/G} \right\}^{T}} & \underset{1 \times n}{\left\{ {- U} \right\}^{T}} \end{Bmatrix}\begin{Bmatrix} \underset{n \times 1}{\left\{ U^{*} \right\}} \\ \underset{n \times 1}{\left\{ {F^{*}/G} \right\}} \end{Bmatrix}}} & (250) \end{matrix}$

The nodal force F is divided into a known part F_(b) and an unknown part F_(v), and the nodal displacement U is divided into a known part U_(b) and unknown part U_(v). Accordingly, the dual nodal displacement U* is divided into an unknown part U_(v)* and a known part U_(b*), and the dual nodal force F* is divided into an unknown part F_(v*) and a known part F_(b*). Consequently, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 110} \right\rbrack & \; \\ {R \equiv {\begin{Bmatrix} \underset{1 \times n_{F_{b}}}{\left\{ {F_{b}/G} \right\}^{T}} & \underset{1 \times n_{F_{v}}}{\left\{ {F_{v}/G} \right\}^{T}} & \underset{1 \times n_{U_{b}}}{\left\{ {- U_{b}} \right\}^{T}} & \underset{1 \times n_{U_{v}}}{\left\{ {- U_{v}} \right\}^{T}} \end{Bmatrix}\begin{Bmatrix} \underset{n_{U_{v}^{*}} \times 1}{\left\{ U_{v}^{*} \right\}} \\ \underset{n_{U_{b}^{*}} \times 1}{\left\{ U_{b}^{*} \right\}} \\ \underset{n_{F_{v}^{*}} \times 1}{\left\{ {F_{v}^{*}/G} \right\}} \\ \underset{n_{F_{b}^{*}} \times 1}{\left\{ {F_{b}^{*}/G} \right\}} \end{Bmatrix}}} & (251) \end{matrix}$

The respective numbers of the parts are as follows:

n _(U) _(v) _(*) =n _(F) _(b)

n _(U) _(b) _(*) =n _(F) _(v)

n _(F) _(v) _(*) =n _(U) _(b)

n _(F) _(b) _(*) =n _(U) _(v)   (252)

These satisfy:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 111} \right\rbrack & \; \\ \begin{matrix} {n = {n_{F_{b}} + n_{F_{v}}}} \\ {= {n_{U_{v}^{*}} + n_{U_{b}^{*}}}} \\ {= {n_{U_{b}} + n_{U_{v}\;}}} \\ {= {n_{F_{v}^{*}} + n_{F_{b}^{*}}}} \end{matrix} & (253) \end{matrix}$

If the known parts F_(b) and U_(b) are replaced with zero, this means that the homogeneous boundary condition is given. Here, with such a condition that the boundary term R is zero, the homogeneous adjoint boundary condition is obtained. Consequently, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 112} \right\rbrack & \; \\ {\underset{n_{U_{b}^{* \times 1}}}{\left\{ U_{b}^{*} \right\}} = \underset{n_{U_{b}^{*}} \times 1}{\left\{ 0 \right\}}} & (254) \\ {\underset{n_{F_{b}^{*}} \times 1}{\left\{ {F_{b}^{*}/G} \right\}} = \underset{n_{F_{b}^{*}} \times 1}{\left\{ 0 \right\}}} & (255) \end{matrix}$

In the case where the combination of the known parts F_(b), U_(b) and the combination of the known parts F_(b*), U_(b*) coincide, this is a self-adjoint boundary condition, and in the case where they do not coincide, this is a non-self-adjoint boundary condition.

8.12 Primal Trial Function and Primal Eigenfunction

The definition formula (216) of the equivalent nodal force is transformed to the following equation:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 113} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n}{\left\lbrack K_{U} \right\rbrack} & \underset{n \times n}{\left\lbrack {- I} \right\rbrack} & \underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n \times 1}{\left\{ U \right\}} \\ \underset{n \times 1}{\left\{ {F/G} \right\}} \\ \underset{\lambda \times 1}{\left\{ c_{o} \right\}} \end{Bmatrix}} = \underset{n \times 1}{\left\{ 0 \right\}}} & (256) \end{matrix}$

where [l] is a unit matrix. This equation represents the relationship that should be established among the nodal displacement U, the nodal force F, and the coefficient C_(o). By combining the internal displacement of the element expressed by eq. (173) with eq. (256), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 114} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n}{\left\lbrack K_{U} \right\rbrack} & \underset{n \times n}{\left\lbrack {- I} \right\rbrack} & \underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack} \\ \underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} & \underset{2 \times n}{\lbrack 0\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n \times 1}{\left\{ U \right\}} \\ \underset{n \times 1}{\left\{ {F/G} \right\}} \\ \underset{\lambda \times 1}{\left\{ c_{o} \right\}} \end{Bmatrix}} = \begin{Bmatrix} \underset{n \times 1}{\left\{ 0 \right\}} \\ \underset{2 \times 1}{\left\{ u \right\}} \end{Bmatrix}} & (257) \end{matrix}$

By combining the known part U_(b) of the nodal displacement U and the known part F_(b) of the nodal force F, we define the nodal known part s_(b) as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 115} \right\rbrack & \; \\ {\underset{n_{b} \times 1}{\left\{ s_{b} \right\}} \equiv \begin{Bmatrix} \underset{n_{U_{b}} \times 1}{\left\{ U_{b} \right\}} \\ \underset{n_{F_{b}} \times 1}{\left\{ {F_{b}/G} \right\}} \end{Bmatrix}} & (258) \end{matrix}$

The number n_(b) of the same is given as:

[Formula 116]

n _(b) ≡n _(U) _(b) +n _(F) _(b)   (259)

By combining the unknown part U_(v) of the nodal displacement U and the unknown part F_(v) of the nodal force F, we define the nodal unknown part s_(v) as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 117} \right\rbrack & \; \\ {\underset{n_{v} \times 1}{\left\{ s_{v} \right\}} \equiv \begin{Bmatrix} \underset{n_{U_{v}} \times 1}{\left\{ U_{v} \right\}} \\ \underset{n_{F_{v}} \times 1}{\left\{ {F_{v}/G} \right\}} \end{Bmatrix}} & (260) \end{matrix}$

The number n_(v) of the same is given as:

[Formula 118]

n _(v) ≡n _(U) _(v) +n _(F) _(v)   (261)

Further, according to eqs. (253), (259), and (261), the following is given:

[Formula 119]

n _(b) +n _(v)=2n  (262)

When U and F in eq. (257) are rearranged in the order according to eqs. (258) and (260), the columns of the matrix parts are also rearranged as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 120} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack} & \underset{n \times n_{v}}{\left\lbrack K_{v} \right\rbrack} & \underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack} \\ \underset{2 \times n_{b}}{\left\lbrack Y_{b} \right\rbrack} & \underset{2 \times n_{v}}{\left\lbrack Y_{v} \right\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n_{b} \times 1}{\left\{ s_{b} \right\}} \\ \underset{n_{v} \times 1}{\left\{ s_{v} \right\}} \\ \underset{\lambda \times 1}{\left\{ c_{o} \right\}} \end{Bmatrix}} = \begin{Bmatrix} \underset{n \times 1}{\left\{ 0 \right\}} \\ \underset{2 \times 1}{\left\{ u \right\}} \end{Bmatrix}} & (263) \end{matrix}$

Here, the matrix [K_(b)] is obtained by extracting corresponding columns from the matrices [K_(U)] and [−I] in the order in which the known parts U_(b) and F_(b) are arranged, and similarly, the matrix [K_(v)] is obtained by extracting corresponding columns from the matrix [K_(u)] and [−I] in the order in which the unknown parts U_(v) and F_(v) are arranged. Further, the matrices [Y_(b)] and [Y_(v)] are obtained in the following manner. With use of the unit matrix [I] and the zero matrix [O], the matrices [χ_(A)] and [0] of eq. (257) are recognized as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 121} \right\rbrack & \; \\ {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n}{\lbrack I\rbrack}}} & (264) \\ {\underset{2 \times n}{\lbrack 0\rbrack} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n}{\lbrack O\rbrack}}} & (265) \end{matrix}$

Then, the operation of extracting columns from the matrix [χ_(A)] can be expressed as an operation of picking out columns from the matrix [I], and similarly, the operation of extracting columns from the matrix [0] can be expressed as an operation of picking out columns of the matrix [O]. The corresponding columns are extracted from the matrix [I] in the order in which the known part U_(b) and the unknown part U_(v) are arranged, thereby composing the matrices [I_(Ub)] and [I_(Uv)], respectively. Likewise, the corresponding columns are extracted from [O] in the order in which the known part F_(b) and the unknown part F_(v) are arranged, thereby composing the matrices [O_(Fb)] and [O_(Fv)], respectively.

Regarding eq. (264), we define:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 122} \right\rbrack & \; \\ {\underset{2 \times n_{U_{b}}}{\left\lbrack \chi_{A_{B}} \right\rbrack} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{U_{b}}}{\left\lbrack I_{U_{b}} \right\rbrack}}} & (266) \\ {\underset{2 \times n_{U_{v}}}{\left\lbrack \chi_{A_{v}} \right\rbrack} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{U_{v}}}{\left\lbrack I_{U_{v}} \right\rbrack}}} & (267) \end{matrix}$

Regarding eq. (265), we define:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 123} \right\rbrack & \; \\ {\underset{2 \times n_{F_{b}}}{\lbrack 0\rbrack} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{F_{b}}}{\left\lbrack O_{F_{b}} \right\rbrack}}} & (268) \\ {\underset{2 \times n_{F_{v}}}{\lbrack 0\rbrack} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{F_{v}}}{\left\lbrack O_{F_{v}} \right\rbrack}}} & (269) \end{matrix}$

Further, we define coefficient matrices as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 124} \right\rbrack & \; \\ {\underset{n \times n_{b}}{\left\lbrack c_{b} \right\rbrack} \equiv \begin{bmatrix} \underset{n \times n_{U_{b}}}{\left\lbrack I_{U_{b}} \right\rbrack} & \underset{n \times n_{F_{b}}}{\left\lbrack O_{F_{b}} \right\rbrack} \end{bmatrix}} & (270) \\ {\underset{n \times n_{v}}{\left\lbrack c_{v} \right\rbrack} \equiv \begin{bmatrix} \underset{n \times n_{U_{v}}}{\left\lbrack I_{U_{v}} \right\rbrack} & \underset{n \times n_{F_{v}}}{\left\lbrack O_{F_{v}} \right\rbrack} \end{bmatrix}} & (271) \end{matrix}$

Then, we obtain the matrices [Y_(b)] and [Y_(v)] as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 125} \right\rbrack & \; \\ {{\underset{2 \times n_{b}}{\left\lbrack Y_{b} \right\rbrack} \equiv {\begin{bmatrix} \underset{2 \times n_{U_{b}}}{\left\lbrack \chi_{A_{b}} \right\rbrack} & \underset{2 \times n_{F_{b}}}{\lbrack 0\rbrack} \end{bmatrix}{\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\begin{bmatrix} \underset{n \times n_{U_{b}}}{\left\lbrack I_{U_{b}} \right\rbrack} & \underset{n \times n_{F_{b}}}{\left\lbrack O_{F_{b}} \right\rbrack} \end{bmatrix}}}} = {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{b}}{\left\lbrack c_{b} \right\rbrack}}} & (272) \\ {{\underset{2 \times n_{v}}{\left\lbrack Y_{v} \right\rbrack} \equiv \begin{bmatrix} \underset{2 \times n_{U_{v}}}{\left\lbrack \chi_{A_{b}} \right\rbrack} & \underset{n \times n_{F_{v}}}{\lbrack 0\rbrack} \end{bmatrix}} = {{\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\begin{bmatrix} \underset{n \times n_{U_{v}}}{\left\lbrack I_{U_{v}} \right\rbrack} & \underset{n \times n_{F_{v}}}{\left\lbrack O_{F_{v}} \right\rbrack} \end{bmatrix}} = {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{v}}{\left\lbrack c_{v} \right\rbrack}}}} & (273) \end{matrix}$

Through the above-described operation, the matrices of eq. (263) are settled, and the displacement u inside the element is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 126} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u \right\}} = {{\underset{2 \times n_{b}}{\left\lbrack Y_{b} \right\rbrack}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}} + {\underset{2 \times n_{v}}{\left\lbrack Y_{v} \right\rbrack}\underset{n_{v} \times 1}{\left\{ s_{v} \right\}}} + {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}}} & (274) \end{matrix}$

In this equation, the displacement u is expressed with the nodal known part s_(b), the nodal unknown part s_(v), and the unknown coefficient c_(o). The upper part of eq. (263) indicates that the following relationship is established between the unknown parts s_(v) and c_(o) and the known part s_(b):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 127} \right\rbrack & \; \\ {{{\underset{n \times n_{v}}{\left\lbrack K_{v} \right\rbrack}\underset{n_{v} \times 1}{\left\{ s_{v} \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} = {{- \underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack}}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}}} & (275) \end{matrix}$

Therefore, we find that the unknown parts s_(v) and c_(o) can be organized better, and the total number of the unknown parts can be reduced.

Then, the unknown parts s_(v) and c_(o) are rearranged so as to constitute unknown parts s_(e) and s_(h), and in accordance with this, the columns of the matrices [K_(v)] and [K₀] are replaced so as to constitute matrices [K_(e)] and [K_(h)]. Likewise, columns of the matrices [Y_(v)] and [ψ₀] are replaced so as to constitute matrices [Y_(e)] and [Y_(h)]. As a result, eq. (263) is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 128} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack} & \underset{n \times n}{\left\lbrack K_{e} \right\rbrack} & \underset{n \times n_{h}}{\left\lbrack K_{h} \right\rbrack} \\ \underset{2 \times n_{b}}{\left\lbrack Y_{b} \right\rbrack} & \underset{2 \times n}{\left\lbrack Y_{e} \right\rbrack} & \underset{n \times n_{h}}{\left\lbrack Y_{h} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n_{b} \times 1}{\left\{ s_{b} \right\}} \\ \underset{n \times 1}{\left\{ s_{e} \right\}} \\ \underset{n_{h} \times 1}{\left\{ s_{h} \right\}} \end{Bmatrix}} = \begin{Bmatrix} \underset{n \times 1}{\left\{ 0 \right\}} \\ \underset{2 \times 1}{\left\{ u \right\}} \end{Bmatrix}} & (276) \end{matrix}$

The purpose of this procedure is to cause [K_(e)] to have an inverse matrix, thereby to eliminate the unknown part s_(e). Therefore, [K_(e)] is a square matrix. According to eqs. (263) and (276), the number n_(h) of the unknown part s_(h) is given as:

[Formula 129]

n _(h) =λ−n+n _(v)  (277)

According to eq. (262), the number may also be given as:

[Formula 130]

n _(h) =λ+n−n _(b)  (278)

The unknown parts s_(e) and s_(h) are composed through the following steps.

(1) Among the unknown part s_(v), unknown variables which are categorized as unknown part s_(e), which are n_(ve) in number, are collected in the first half of the unknown part s_(e).

(2) Among the unknown part c_(o), unknown variables which are categorized as unknown part s_(e), which are n_(oe) in number, are collected in the latter half of the unknown part s_(e).

(3) Among the unknown part s_(v), unknown variables which are categorized as unknown part s_(h), which are n_(vh) in number, are collected in the first half of the unknown part s_(h).

(4) Among the unknown part c_(o), unknown variables which are categorized as unknown part s_(h), which are n_(oh) in number, are collected in the latter half of the unknown part s_(h).

In this way, the number of the unknown part sn is given as:

[Formula 131]

n _(h) =n _(vh) +n _(oh)  (279)

According to the steps (1) and (3), we obtain:

[Formula 132]

n _(v) =n _(ve) +n _(vh)  (280)

According to the steps (2) and (4), we obtain:

[Formula 133]

λ=n _(oe) +n _(oh)  (281)

According to eqs. (277), (279), (280), and (281), we obtain:

[Formula 134]

n _(ve) +n _(oe) =n  (282)

This indicates the number of the unknown part s_(e) is n.

In the self-adjoint boundary condition, the following is satisfied:

[Formula 135]

n _(b) =n _(v) =n  (283)

Therefore, by substituting this equation into eq. (277) or eq. (278), we obtain:

[Formula 136]

n _(h)=λ  (284)

Further, according to eqs. (279), (281), and (284), we obtain:

[Formula 137]

n _(vh) =n _(oe)  (285)

Since the matrix [Y_(v)] is expressed by eq. (273), the operation of extracting columns from the matrix [Y_(v)] can be expressed by an operation of picking out columns of the matrix [c_(v)]. By extracting the columns from the matrix [c_(v)] to compose [c_(ve)] and [c_(vh)], we define:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 138} \right\rbrack & \; \\ {\underset{2 \times n_{ve}}{\left\lbrack Y_{ve} \right\rbrack} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{ve}}{\left\lbrack c_{ve} \right\rbrack}}} & (286) \\ {\underset{2 \times n_{vh}}{\left\lbrack Y_{vh} \right\rbrack} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{vh}}{\left\lbrack c_{vh} \right\rbrack}}} & (287) \end{matrix}$

With use of the unit matrix [I], the matrix [ψ₀] of eq. (263) is recognized as:

$\begin{matrix} {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \equiv {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times \lambda}{\lbrack I\rbrack}}} & (288) \end{matrix}$

Then, the operation of extracting columns from the matrix [ψ₀] can be expressed as an operation of picking out columns from the matrix [I]. By extracting the columns from the matrix [I] to compose [I_(oe)] and [I_(oh)], we define:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 139} \right\rbrack & \; \\ {\underset{2 \times n_{oe}}{\left\lbrack \psi_{oe} \right\rbrack} \equiv {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times n_{oe}}{\left\lbrack I_{oe} \right\rbrack}}} & (289) \\ {\underset{2 \times n_{oh}}{\left\lbrack \psi_{oh} \right\rbrack} \equiv {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times n_{oh}}{\left\lbrack I_{oh} \right\rbrack}}} & (290) \end{matrix}$

Therefore, the matrices [Y_(e)], [Y_(h)] are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 141} \right\rbrack & \; \\ {{\underset{2 \times n}{\left\lbrack Y_{e} \right\rbrack} \equiv \begin{bmatrix} \underset{2 \times n_{ve}}{\left\lbrack Y_{ve} \right\rbrack} & \underset{2 \times n_{oe}}{\left\lbrack \psi_{oe} \right\rbrack} \end{bmatrix}} = {\begin{bmatrix} \underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}\begin{bmatrix} \underset{n \times n_{ve}}{\left\lbrack c_{ve} \right\rbrack} & \underset{n \times n_{oe}}{\lbrack 0\rbrack} \\ \underset{\lambda \times n_{ve}}{\lbrack 0\rbrack} & \underset{\lambda \times n_{oe}}{\left\lbrack I_{oe} \right\rbrack} \end{bmatrix}}} & (291) \\ {{\underset{2 \times n_{h}}{\left\lbrack Y_{h} \right\rbrack} \equiv \begin{bmatrix} \underset{2 \times n_{vh}}{\left\lbrack Y_{vh} \right\rbrack} & \underset{2 \times n_{oh}}{\left\lbrack \psi_{oh} \right\rbrack} \end{bmatrix}} = {\begin{bmatrix} \underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}\begin{bmatrix} \underset{n \times n_{vh}}{\left\lbrack c_{vh} \right\rbrack} & \underset{n \times n_{oh}}{\lbrack 0\rbrack} \\ \underset{\lambda \times n_{vh}}{\lbrack 0\rbrack} & \underset{\lambda \times n_{oh}}{\left\lbrack I_{oh} \right\rbrack} \end{bmatrix}}} & (292) \end{matrix}$

Further, we define a function matrix as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 142} \right\rbrack & \; \\ {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack} \equiv \left\lbrack {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \right\rbrack} & (293) \end{matrix}$

We also define coefficient matrices as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 143} \right\rbrack & \; \\ {\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e} \right\rbrack}\begin{bmatrix} \underset{n \times n_{ve}}{\left\lbrack c_{ve} \right\rbrack} & \underset{n \times n_{oe}}{\lbrack 0\rbrack} \\ \underset{\lambda \times n_{ve}}{\lbrack 0\rbrack} & \underset{\lambda \times n_{oe}}{\left\lbrack I_{oe} \right\rbrack} \end{bmatrix}} & (294) \\ {\underset{{({n + \lambda})} \times n_{h}}{\left\lbrack c_{h} \right\rbrack}\begin{bmatrix} \underset{n \times n_{vh}}{\left\lbrack c_{vh} \right\rbrack} & \underset{n \times n_{oh}}{\lbrack 0\rbrack} \\ \underset{\lambda \times n_{vh}}{\lbrack 0\rbrack} & \underset{\lambda \times n_{oh}}{\left\lbrack I_{oh} \right\rbrack} \end{bmatrix}} & (295) \end{matrix}$

Then, the matrices [Y_(e)] and [Y_(h)] are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 144} \right\rbrack & \; \\ {\underset{2 \times n}{\left\lbrack Y_{e} \right\rbrack} \equiv {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e} \right\rbrack}}} & (296) \\ {\underset{2 \times n_{h}}{\left\lbrack Y_{h} \right\rbrack} \equiv {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n_{h}}{\left\lbrack c_{h} \right\rbrack}}} & (297) \end{matrix}$

When eq. (276) is separated, they are transformed to the following equations:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 145} \right\rbrack & \; \\ {{\underset{n \times n}{\left\lbrack K_{e} \right\rbrack}\underset{n \times 1}{\left\{ s_{e} \right\}}} = {{- \underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack}}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}\underset{n \times n_{h}}{\left\lbrack K_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ s_{h} \right\}}}} & (298) \\ {\underset{2 \times 1}{\left\{ u \right\}} = {{\underset{2 \times n_{b}}{\left\lbrack Y_{b} \right\rbrack}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}} + {\underset{2 \times n}{\left\lbrack Y_{e} \right\rbrack}\underset{n \times 1}{\left\{ s_{e} \right\}}} + {\underset{2 \times n_{h}}{\left\lbrack Y_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ s_{h} \right\}}}}} & (299) \end{matrix}$

By solving the unknown part s_(e) of eq. (298), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 146} \right\rbrack & \; \\ {\underset{n \times 1}{\left\{ s_{e} \right\}} = {{{- {\underset{n \times n}{\left\lbrack K_{e} \right\rbrack}}^{- 1}}\underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}} - {{\underset{n \times n}{\left\lbrack K_{e} \right\rbrack}}^{- 1}\underset{n \times n_{h}}{\left\lbrack K_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ s_{h} \right\}}}}} & (300) \end{matrix}$

By substituting this equation into eq. (299), we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 147} \right\rbrack} & \; \\ {\underset{2 \times 1}{\left\{ u \right\}} = {{\left( {\underset{2 \times n_{b}}{\left\lbrack Y_{b} \right\rbrack} - {\underset{2 \times n}{\left\lbrack Y_{e} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e} \right\rbrack}}^{- 1}\underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack}}} \right)\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}} + {\left( {\underset{2 \times n_{h}}{\left\lbrack Y_{h} \right\rbrack} - {\underset{2 \times n}{\left\lbrack Y_{e} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e} \right\rbrack}}^{- 1}\underset{n \times n_{h}}{\left\lbrack K_{h} \right\rbrack}}} \right)\underset{n_{h} \times 1}{\left\{ s_{h} \right\}}}}} & (301) \end{matrix}$

The first term is a function having the known part s_(b), and becomes a displacement {u_(B)} that satisfies the inhomogeneous boundary condition. The second term is a function having the unknown part s_(h), and becomes a displacement {u_(H)} that satisfies the homogeneous boundary condition. In other words, we define:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 148} \right\rbrack & \; \\ {\underset{2 \times n_{b}}{\left\lbrack \chi_{B} \right\rbrack} \equiv {\underset{2 \times n_{b}}{\left\lbrack Y_{b} \right\rbrack} - {\underset{2 \times n}{\left\lbrack Y_{e} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e} \right\rbrack}}^{- 1}\underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack}}}} & (302) \\ {\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack} \equiv {\underset{2 \times n_{h}}{\left\lbrack Y_{h} \right\rbrack} - {\underset{2 \times n}{\left\lbrack Y_{e} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e} \right\rbrack}}^{- 1}\underset{n \times n_{h}}{\left\lbrack K_{h} \right\rbrack}}}} & (303) \end{matrix}$

Then, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 149} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u_{B} \right\}} \equiv {\underset{2 \times n_{b}}{\left\lbrack \chi_{B} \right\rbrack}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}}} & (304) \\ {\underset{2 \times 1}{\left\{ u_{H} \right\}} \equiv {\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ s_{h} \right\}}}} & (305) \end{matrix}$

And we find:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 150} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u \right\}} \equiv {\underset{2 \times 1}{\left\{ u_{B} \right\} +}\underset{2 \times 1}{\left\{ u_{H} \right\}}}} & (306) \end{matrix}$

This equation shows details of eq. (174), and we find that the displacement {u} is expressed with the displacement {u_(B)} that satisfies the inhomogeneous boundary condition and the displacement {u_(H)} that satisfies the homogeneous boundary condition.

According to eqs. (272), (296), and (302), the matrix [χ_(B)] is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 151} \right\rbrack & \; \\ {\underset{2 \times n_{b}}{\left\lbrack \chi_{B} \right\rbrack} \equiv {{\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{b}}{\left\lbrack c_{b} \right\rbrack}} - {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e} \right\rbrack}\underset{n \times n}{\left\lbrack K_{e} \right\rbrack^{- 1}}\underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack}}}} & (307) \end{matrix}$

According to eq. (293), we find that the first term of this equation is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 152} \right\rbrack & \; \\ {{\underset{2 \times n_{b}}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{b}}{\left\lbrack c_{b} \right\rbrack}} = {{\begin{bmatrix} \underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}\begin{bmatrix} \underset{n \times n_{b}}{\left\lbrack c_{b} \right\rbrack} \\ \underset{\lambda \times n_{b}}{\lbrack 0\rbrack} \end{bmatrix}} = {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\begin{bmatrix} \underset{n \times n_{b}}{\left\lbrack c_{b} \right\rbrack} \\ \underset{\lambda \times n_{b}}{\lbrack 0\rbrack} \end{bmatrix}}}} & (308) \end{matrix}$

Focusing this, we can give the following:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 153} \right\rbrack & \; \\ {\underset{{({n + \lambda})} \times n_{b}}{\left\lbrack H_{b} \right\rbrack} \equiv {\begin{bmatrix} \underset{n \times n_{b}}{\left\lbrack c_{b} \right\rbrack} \\ \underset{\lambda \times n_{b}}{\lbrack 0\rbrack} \end{bmatrix} - {\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e} \right\rbrack}}^{- 1}\underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack}}}} & (309) \end{matrix}$

Then, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 154} \right\rbrack & \; \\ {\underset{2 \times n_{b}}{\left\lbrack \chi_{B} \right\rbrack} \equiv {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n_{b}}{\left\lbrack H_{b} \right\rbrack}}} & (310) \end{matrix}$

According to eqs. (296), (297), and (303), the matrix [ψ_(h)] is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 155} \right\rbrack & \; \\ {\underset{{({n + \lambda})} \times n_{b}}{\left\lbrack \psi_{h} \right\rbrack} \equiv {{\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n_{h}}{\left\lbrack c_{h} \right\rbrack}} - {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e} \right\rbrack}}^{- 1}\underset{n \times n_{h}}{\left\lbrack K_{h} \right\rbrack}}}} & (311) \end{matrix}$

The following is assumed:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 156} \right\rbrack & \; \\ {\underset{{({n + \lambda})} \times n_{h}}{\left\lbrack H_{h} \right\rbrack} \equiv {\underset{{({n + \lambda})} \times n_{h}}{\left\lbrack c_{h} \right\rbrack} - {\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e} \right\rbrack}}^{- 1}\underset{n \times n_{h\;}}{\left\lbrack K_{h} \right\rbrack}}}} & (312) \end{matrix}$

Then, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 157} \right\rbrack & \; \\ {\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack} \equiv {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n_{h}}{\left\lbrack H_{h} \right\rbrack}}} & (313) \end{matrix}$

According to eqs. (310) and (313), we find that the matrices [χ_(B)] and [ψ_(h)] are separated into a function part [Γ] and coefficient parts [H_(b)] and [H_(h)]. Further, the displacement {u} is expressed as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 158} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u \right\}} = {{\underset{2 \times 1}{\left\{ u_{B} \right\}} + \underset{2 \times 1}{\left\{ u_{H} \right\}}} = {{\underset{2 \times n_{b}}{\left\lbrack \chi_{B} \right\rbrack}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}} + {\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ s_{h} \right\}}}}}} & (314) \end{matrix}$

From this, we find that an arbitrary nodal known part s_(b) composes a function that satisfies the inhomogeneous boundary condition, and an arbitrary unknown part s_(h) composes a function that satisfies the homogeneous boundary condition. That the displacement {u_(H)} satisfies the homogeneous boundary condition with respect to the unknown part s_(h) is achieved by the effects of the matrix [ψ_(h)]. Therefore, let the coefficient vector be {e_(h)}, and we can give the primal eigenfunction {φ} as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 159} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ \varphi \right\}} \equiv {\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ e_{h} \right\}}}} & (315) \end{matrix}$

At the same time, we can give the variation {δφ} as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 160} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ {\delta \; \varphi} \right\}} \equiv {\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ {\delta \; e_{h}} \right\}}}} & (316) \end{matrix}$

The matrix [ψ_(h)], which plays an important role, is referred to as a “primal trial function”.

8.13 Dual Trial Function and Dual Eigenfunction

Transforming the definition formula (235) of the equivalent nodal force, we obtain the following equation:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 161} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n}{\left\lbrack K_{U} \right\rbrack} & \underset{n \times n}{\left\lbrack {- I} \right\rbrack} & \underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n \times 1}{\left\{ U^{*} \right\}} \\ \underset{n \times 1}{\left\{ {F^{*}/G} \right\}} \\ \underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}} \end{Bmatrix}} = \underset{n \times 1}{\left\{ 0 \right\}}} & (317) \end{matrix}$

This equation represents the relationship that should be established among the dual nodal displacement U*, the dual nodal force F*, and the coefficient c_(o)*. By combining an internal displacement u*of the element expressed by eq. (190) into eq. (317), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 162} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n}{\left\lbrack K_{U} \right\rbrack} & \underset{n \times n}{\left\lbrack {- I} \right\rbrack} & \underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack} \\ \underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} & \underset{2 \times n}{\lbrack 0\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n \times 1}{\left\{ U^{*} \right\}} \\ \underset{n \times 1}{\left\{ {F^{*}/G} \right\}} \\ \underset{{\lambda \times 1}\;}{\left\{ c_{o}^{*} \right\}} \end{Bmatrix}} = \begin{Bmatrix} \underset{n \times 1}{\left\{ 0 \right\}} \\ \underset{2 \times 1}{\left\{ u^{*} \right\}} \end{Bmatrix}} & (318) \end{matrix}$

By combining the known part U_(b)* of the dual nodal displacement U* and the known part F_(b)* of the dual nodal force F*, we define the nodal known part s_(b)* as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 163} \right\rbrack & \; \\ {\underset{n_{b}^{*} \times 1}{\left\{ s_{b}^{*} \right\}} \equiv \begin{Bmatrix} \underset{1 \times n_{U_{b}^{*}}}{\left\{ U_{b}^{*} \right\}} \\ \underset{1 \times n_{F_{b}^{*}}}{\left\{ {F_{b}^{*}\text{/}G} \right\}} \end{Bmatrix}} & (319) \end{matrix}$

The number n_(b)* of the same is given as:

[Formula 164]

n _(b) *≡n _(U) _(b) _(*) +n _(F) _(b) _(*)   (320)

By combining the unknown part U_(v)*of the dual nodal displacement U* and the unknown part F_(v)*of the dual nodal force F*, we define the nodal unknown part s_(v)* as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 165} \right\rbrack & \; \\ {\underset{n_{v}^{*} \times 1}{\left\{ s_{v}^{*} \right\}} \equiv \begin{Bmatrix} \underset{1 \times n_{U_{v}^{*}}}{\left\{ U_{v}^{*} \right\}} \\ \underset{1 \times n_{F_{v}^{*}}}{\left\{ {F_{v}^{*}\text{/}G} \right\}} \end{Bmatrix}} & (321) \end{matrix}$

The number n_(v)* of the same is given as:

[Formula 166]

n _(v) *≡n _(U) _(v) _(*) +n _(F) _(v) _(*)   (322)

Further, according to eqs. (253), (320), and (322), we find:

[Formula 167]

n _(b) *+n _(v)*=2n  (323)

If U and F in eq. (318) are rearranged in the order according to eqs. (319) and (321), the columns of the matrix parts are also rearranged as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 168} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n_{b}^{*}}{\left\lbrack K_{b}^{*} \right\rbrack} & \underset{n \times n_{v}^{*}}{\left\lbrack K_{v}^{*} \right\rbrack} & \underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack} \\ \underset{2 \times n_{b}^{*}}{\left\lbrack Y_{b}^{*} \right\rbrack} & \underset{2 \times n_{v}^{*}}{\left\lbrack Y_{v}^{*} \right\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n_{b}^{*} \times 1}{\left\{ s_{b}^{*} \right\}} \\ \underset{n_{v}^{*} \times 1}{\left\{ s_{v}^{*} \right\}} \\ \underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}} \end{Bmatrix}} = \begin{Bmatrix} \underset{n \times 1}{\left\{ 0 \right\}} \\ \underset{2 \times 1}{\left\{ u^{*} \right\}} \end{Bmatrix}} & (324) \end{matrix}$

Here, the matrix [K_(b)*] is obtained by extracting corresponding columns from the matrices [K_(U)] and [−I] in the order in which the known parts U_(b)* and F_(b)* are arranged, and similarly, the matrix [K_(v)*] is obtained by extracting corresponding columns from the matrix [K_(U)] and [−I] in the order in which the unknown parts U_(v)* and F_(v)* are arranged. Further, the matrices [Y_(b)*] and [Y_(v)*] are obtained in the following manner. With use of the unit matrix [I] and the zero matrix [O], the matrices [χ_(A)] and [0] of eq. (318) are recognized as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 169} \right\rbrack & \; \\ {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack} \equiv {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times n}{\lbrack I\rbrack}}} & (325) \\ {\underset{2 \times n}{\lbrack 0\rbrack} \equiv {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times n}{\lbrack O\rbrack}}} & (326) \end{matrix}$

Then, the operation of extracting columns from the matrix [χ_(A)] can be expressed as an operation of picking out columns from the matrix [I], and similarly, the operation of extracting columns from the matrix [0] can be expressed as an operation of picking out columns of the matrix [O]. The corresponding columns are extracted from the matrix [I] in the order in which the known parts U_(b)* and the unknown parts U_(v)* are arranged, which thereby compose the matrices [I_(Ub*)] and [I_(Uv*)], respectively. Likewise, the corresponding columns are extracted from “O” in the order in which the known parts F_(b)* and the unknown parts F_(v)* are arranged, which thereby compose the matrices [O_(Fb*)] and [O_(Fv*)], respectively.

Regarding eq. (325), we define:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 170} \right\rbrack & \; \\ {\underset{2 \times n_{U_{b}^{*}}}{\left\lbrack _{A_{b}^{*}} \right\rbrack} \equiv {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times n_{U_{b}^{*}}}{\left\lbrack I_{U_{b}^{*}} \right\rbrack}}} & (327) \\ {\underset{2 \times n_{U_{v}^{*}}}{\left\lbrack _{A_{v}^{*}} \right\rbrack} \equiv {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times n_{U_{v}^{*}}}{\left\lbrack I_{U_{v}^{*}} \right\rbrack}}} & (328) \end{matrix}$

Regarding eq. (326), we define:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 171} \right\rbrack & \; \\ {\underset{2 \times n_{F_{b}^{*}}}{\lbrack 0\rbrack} \equiv {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times n_{F_{b}^{*}}}{\left\lbrack O_{F_{b}^{*}} \right\rbrack}}} & (329) \\ {\underset{2 \times n_{F_{v}^{*}}}{\lbrack 0\rbrack} \equiv {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times n_{F_{v}^{*}}}{\left\lbrack O_{F_{v}^{*}} \right\rbrack}}} & (330) \end{matrix}$

Further, we define coefficient matrices as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 172} \right\rbrack & \; \\ {\underset{n \times n_{b}^{*}}{\left\lbrack c_{b}^{*} \right\rbrack} \equiv \left\lbrack {\underset{n \times n_{U_{b}^{*}}}{\left\lbrack I_{U_{b}^{*}} \right\rbrack}\mspace{14mu} \underset{n \times n_{F_{b}^{*}}}{\left\lbrack O_{F_{b}^{*}} \right\rbrack}} \right\rbrack} & (331) \\ {\underset{n \times n_{v}^{*}}{\left\lbrack c_{v}^{*} \right\rbrack} \equiv \left\lbrack {\underset{n \times n_{U_{v}^{*}}}{\left\lbrack I_{U_{v}^{*}} \right\rbrack}\mspace{14mu} \underset{n \times n_{F_{v}^{*}}}{\left\lbrack O_{F_{v}^{*}} \right\rbrack}} \right\rbrack} & (332) \end{matrix}$

Then, we obtain the matrices [Y_(b*)] and [Y_(v*)] as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 173} \right\rbrack & \; \\ {{\underset{2 \times n_{b}^{*}}{\left\lbrack Y_{b}^{*} \right\rbrack} \equiv \left\lbrack {\underset{2 \times n_{U_{b}^{*}}}{\left\lbrack _{A_{b}^{*}} \right\rbrack}\mspace{14mu} \underset{2 \times n_{F_{b}^{*}}}{\lbrack 0\rbrack}} \right\rbrack} = {{\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\left\lbrack {\underset{n \times n_{U_{b}^{*}}}{\left\lbrack I_{U_{b}^{*}} \right\rbrack \mspace{11mu}}\; \underset{n \times n_{F_{b}^{*}}}{\left\lbrack O_{F_{b}^{*}} \right\rbrack}} \right\rbrack} = {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times n_{b}^{*}}{\left\lbrack c_{b}^{*} \right\rbrack}}}} & (333) \\ {{\underset{2 \times n_{v}^{*}}{\left\lbrack Y_{v}^{*} \right\rbrack} \equiv \left\lbrack {\underset{2 \times n_{U_{v}^{*}}}{\left\lbrack _{A_{v}^{*}} \right\rbrack \;}\mspace{11mu} \underset{2 \times n_{F_{v}^{*}}}{\lbrack 0\rbrack}} \right\rbrack} = {{\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\left\lbrack {\underset{n \times n_{U_{v}^{*}}}{\left\lbrack I_{U_{v}^{*}} \right\rbrack \mspace{11mu}}\; \underset{n \times n_{F_{v}^{*}}}{\left\lbrack O_{F_{v}^{*}} \right\rbrack}} \right\rbrack} = {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times n_{v}^{*}}{\left\lbrack c_{v}^{*} \right\rbrack}}}} & (334) \end{matrix}$

Through the above-described operation, the matrices of eq. (324) are settled, and the internal dual displacement u* of the element is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 174} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u^{*} \right\}} = {{\underset{2 \times n_{b}^{*}}{\left\lbrack Y_{b}^{*} \right\rbrack}\underset{n_{b}^{*} \times 1}{\left\{ s_{b}^{*} \right\}}} + {\underset{2 \times n_{v}^{*}}{\left\lbrack Y_{v}^{*} \right\rbrack}\underset{n_{v}^{*} \times 1}{\left\{ s_{v}^{*} \right\}}} + {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}}} & (335) \end{matrix}$

In this equation, the dual displacement u* is expressed with the nodal known part s_(b)*, the nodal unknown part s_(v)*, and the unknown coefficient c_(o)*. The upper part of eq. (324) indicates that the following relationship is established between the unknown parts s_(v)*, c_(o)* and the known part s_(b)*:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 175} \right\rbrack & \; \\ {{{\underset{n \times n_{v}^{*}}{\left\lbrack K_{v}^{*} \right\rbrack}\underset{n_{v}^{*} \times 1}{\left\{ s_{v}^{*} \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} = {{- \underset{n \times n_{b}^{*}}{\left\lbrack K_{b}^{*} \right\rbrack}}\underset{n_{b}^{*} \times 1}{\left\{ s_{b}^{*} \right\}}}} & (336) \end{matrix}$

Therefore, we find that the unknown parts s_(v)* and c_(o)* can be organized better, and the total number of the unknown parts can be reduced.

Then, the unknown parts s_(v)* and c_(o)* are rearranged so as to constitute unknown parts s_(e)* and s_(b)*, and in accordance with this, the columns of the matrices [K_(v*)] and [K₀] are replaced so as to constitute matrices [K_(e*)] and [K_(h*)]. Likewise, columns of the matrices [Y_(v*)] and [ψ₀] are replaced so as to constitute matrices [Y_(e*)] and [Y_(h*)]. As a result, eq. (324) is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 176} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n_{b}^{*}}{\left\lbrack K_{b}^{*} \right\rbrack} & \underset{n \times n}{\left\lbrack K_{e}^{*} \right\rbrack} & \underset{n \times n_{h}^{*}}{\left\lbrack K_{h}^{*} \right\rbrack} \\ \underset{2 \times n_{b}^{*}}{\left\lbrack Y_{b}^{*} \right\rbrack} & \underset{2 \times n}{\left\lbrack Y_{e}^{*} \right\rbrack} & \underset{2 \times n_{h}^{*}}{\left\lbrack Y_{h}^{*} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n_{b}^{*} \times 1}{\left\{ s_{b}^{*} \right\}} \\ \underset{n \times 1}{\left\{ s_{e}^{*} \right\}} \\ \underset{n_{h}^{*} \times 1}{\left\{ s_{h}^{*} \right\}} \end{Bmatrix}} = \begin{Bmatrix} \underset{n \times 1}{\left\{ 0 \right\}} \\ \underset{2 \times 1}{\left\{ u^{*} \right\}} \end{Bmatrix}} & (337) \end{matrix}$

The purpose of this procedure is to cause [K_(e)*] to have an inverse matrix, thereby to eliminate the unknown part s_(e)*. Therefore, [K_(e*)] is a square matrix. According to eqs. (324) and (337), the number n_(h)* of the unknown part s_(h)* is given as:

[Formula 177]

n _(h) *=λ−n+n _(v)*  (338)

According to eq. (323), the number may also be given as:

[Formula 178]

n _(h) *=λ+n−n _(b)*  (339)

The unknown parts s_(e)* and s_(h)* are composed through the following steps.

(1) Among the unknown part s_(v)*, unknown variables which are categorized as unknown part s_(e)*, which are n_(ve)* in number, are collected in the first half of the unknown part s_(e)*.

(2) Among the unknown part c_(o)*, unknown variables which are categorized as unknown part s_(e)*, which are n_(oe)* in number, are collected in the latter half of the unknown part s_(e)*.

(3) Among the unknown part s_(v)*, unknown variables which are categorized as unknown part s_(h)*, which are n_(vn)* in number, are collected in the first half of the unknown part s_(h)*.

(4) Among the unknown part c_(o)*, unknown variables which are categorized as unknown part s_(h)*, which are n_(oh)* in number, are collected in the latter half of the unknown part s_(h)*.

In this way, the number of the unknown part s_(h) is given as:

[Formula 179]

n _(h) *=n _(vh) *+n _(oh)*  (340)

According to the steps (1) and (3), we obtain:

[Formula 180]

n _(v) *=n _(ve) *+n _(vh)*  (341)

According to the steps (2) and (4), we obtain:

[Formula 181]

λ=n _(oe) *+n _(oh)*  (342)

According to eqs. (338), (340), (341), and (342), we obtain:

[Formula 182]

n _(ve) *+n _(oe) *=n  (343)

This indicates the number of the unknown part s_(e)* is n.

In the self-adjoint boundary condition, the following is satisfied:

[Formula 183]

n _(b) *=n _(v) *=n  (344)

Therefore, by substituting this equation into eq. (338) or eq. (339), we obtain:

[Formula 184]

n _(h)*=λ  (345)

Further, according to eqs. (340), (342), and (345), we obtain:

[Formula 185]

n _(vh) *=n _(oe)*  (346)

Since the matrix [Y_(v)*] is expressed by eq. (334), the operation of extracting columns from the matrix [Y_(v)*] can be expressed by an operation of picking out columns of the matrix [c_(v)*]. By extracting the columns from the matrix [c_(v)*] to compose [c_(ve)*] and [c_(vh)*], we define:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 186} \right\rbrack & \; \\ {\underset{2 \times n_{ve}^{*}}{\left\lbrack Y_{ve}^{*} \right\rbrack} \equiv {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times n_{ve}^{*}}{\left\lbrack c_{ve}^{*} \right\rbrack}}} & (347) \\ {\underset{2 \times n_{vh}^{*}}{\left\lbrack Y_{vh}^{*} \right\rbrack} \equiv {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\underset{n \times n_{vh}^{*}}{\left\lbrack c_{vh}^{*} \right\rbrack}}} & (348) \end{matrix}$

With use of the unit matrix [I], the matrix [ψ₀] of eq. (324) is recognized as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 187} \right\rbrack & \; \\ {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \equiv {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times \lambda}{\lbrack I\rbrack}}} & (349) \end{matrix}$

Then, the operation of extracting columns from the matrix [ψ₀] can be expressed as an operation of picking out columns from the matrix [I]. By extracting the columns from the matrix [I] to compose [I_(oe)*] and [I_(oh)*], we define:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 188} \right\rbrack & \; \\ {\underset{2 \times n_{oe}^{*}}{\left\lbrack \psi_{oe}^{*} \right\rbrack} \equiv {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times n_{oe}^{*}}{\left\lbrack I_{oe}^{*} \right\rbrack}}} & (350) \\ {\underset{2 \times n_{oh}^{*}}{\left\lbrack \psi_{oh}^{*} \right\rbrack} \equiv {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times n_{oh}^{*}}{\left\lbrack I_{oh}^{*} \right\rbrack}}} & (351) \end{matrix}$

Therefore, the matrices [Y_(e)*] and [Y_(h)*] are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 189} \right\rbrack & \; \\ {{\underset{2 \times n}{\left\lbrack Y_{e}^{*} \right\rbrack} \equiv \left\lbrack {\underset{2 \times n_{ve}^{*}}{\left\lbrack Y_{ve}^{*} \right\rbrack}\mspace{11mu} \underset{2 \times n_{oe}^{*}}{\; \left\lbrack \psi_{oe}^{*} \right\rbrack}} \right\rbrack} = {\left\lbrack {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack \;}\mspace{11mu} \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \right\rbrack \begin{bmatrix} \underset{n \times n_{ve}^{*}}{\left\lbrack c_{ve}^{*} \right\rbrack} & \underset{n \times n_{oe}^{*}}{\lbrack 0\rbrack} \\ \underset{\lambda \times n_{ve}^{*}}{\lbrack 0\rbrack} & \underset{\lambda \times n_{oe}^{*}}{\left\lbrack I_{oe}^{*} \right\rbrack} \end{bmatrix}}} & (352) \\ {{\underset{2 \times n_{h}^{*}}{\left\lbrack Y_{h}^{*} \right\rbrack} \equiv \left\lbrack {\underset{2 \times n_{vh}^{*}}{\left\lbrack Y_{vh}^{*} \right\rbrack}\mspace{14mu} \underset{2 \times n_{oh}^{*}}{\left\lbrack \psi_{oh}^{*} \right\rbrack}} \right\rbrack} = {\left\lbrack {\underset{2 \times n}{\left\lbrack _{A} \right\rbrack}\mspace{14mu} \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \right\rbrack \begin{bmatrix} \underset{n \times n_{vh}^{*}}{\left\lbrack c_{vh}^{*} \right\rbrack} & \underset{n \times n_{ch}^{*}}{\lbrack 0\rbrack} \\ \underset{\lambda \times n_{vh}^{*}}{\lbrack 0\rbrack} & \underset{\lambda \times n_{oh}^{*}}{\left\lbrack I_{oh}^{*} \right\rbrack} \end{bmatrix}}} & (353) \end{matrix}$

Further, we define coefficient matrices as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 190} \right\rbrack & \; \\ {\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e}^{*} \right\rbrack} \equiv \begin{bmatrix} \underset{n \times n_{ve}^{*}}{\left\lbrack c_{ve}^{*} \right\rbrack} & \underset{n \times n_{oe}^{*}}{\lbrack 0\rbrack} \\ \underset{\lambda \times n_{ve}^{*}}{\lbrack 0\rbrack} & \underset{\lambda \times n_{oe}^{*}}{\left\lbrack I_{oe}^{*} \right\rbrack} \end{bmatrix}} & (354) \\ {\underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack c_{h}^{*} \right\rbrack} \equiv \begin{bmatrix} \underset{n \times n_{vh}^{*}}{\left\lbrack c_{vh}^{*} \right\rbrack} & \underset{n \times n_{oh}^{*}}{\lbrack 0\rbrack} \\ \underset{\lambda \times n_{vh}^{*}}{\lbrack 0\rbrack} & \underset{\lambda \times n_{oh}^{*}}{\left\lbrack I_{oh}^{*} \right\rbrack} \end{bmatrix}} & (355) \end{matrix}$

Then, the matrices [Y_(e)*] and [Y_(h)*] are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 191} \right\rbrack & \; \\ {\underset{2 \times n}{\left\lbrack Y_{e}^{*} \right\rbrack} \equiv {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e}^{*} \right\rbrack}}} & (356) \\ {\underset{2 \times n_{h}^{*}}{\left\lbrack Y_{h}^{*} \right\rbrack} \equiv {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack c_{h}^{*} \right\rbrack}}} & (357) \end{matrix}$

Separating eq. (337), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 192} \right\rbrack & \; \\ {{\underset{n \times n}{\left\lbrack K_{e}^{*} \right\rbrack}\underset{n \times 1}{\left\{ s_{e}^{*} \right\}}} = {{{- \underset{n \times n_{b}^{*}}{\left\lbrack K_{b}^{*} \right\rbrack}}\underset{n_{b}^{*} \times 1}{\left\{ s_{b}^{*} \right\}}} - {\underset{n \times n_{h}^{*}}{\left\lbrack K_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ s_{h}^{*} \right\}}}}} & (358) \\ {\underset{2 \times 1}{\left\{ u^{*} \right\}} = {{\underset{2 \times n_{b}^{*}}{\left\lbrack Y_{b}^{*} \right\rbrack}\underset{n_{b}^{*} \times 1}{\left\{ s_{b}^{*} \right\}}} + {\underset{2 \times n}{\left\lbrack Y_{e}^{*} \right\rbrack}\underset{n \times 1}{\left\{ s_{e}^{*} \right\}}} + {\underset{2 \times n_{h}^{*}}{\left\lbrack Y_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ s_{h}^{*} \right\}}}}} & (359) \end{matrix}$

Solving the unknown part s_(e)* in eq. (358), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 193} \right\rbrack & \; \\ {\underset{n \times 1}{\left\{ s_{e}^{*} \right\}} = {{{- {\underset{n \times n}{\left\lbrack K_{b}^{*} \right\rbrack}}^{- 1}}\underset{n \times n_{b}^{*}}{\left\lbrack K_{b}^{*} \right\rbrack}\underset{n_{b}^{*} \times 1}{\left\{ s_{b}^{*} \right\}}} - {{\underset{n \times n}{\left\lbrack K_{e}^{*} \right\rbrack}}^{- 1}\underset{n \times n_{h}^{*}}{\left\lbrack K_{n}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ s_{h}^{*} \right\}}}}} & (360) \end{matrix}$

By substituting this equation into eq. (359), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 194} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u^{*} \right\}} = {{\left( {\underset{2 \times n_{b}^{*}}{\left\lbrack Y_{b}^{*} \right\rbrack} - {\underset{2 \times n}{\left\lbrack Y_{e}^{*} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e}^{*} \right\rbrack}}^{- 1}\underset{n \times n_{b}^{*}}{\left\lbrack K_{b}^{*} \right\rbrack}}} \right)\underset{n_{b}^{*} \times 1}{\left\{ s_{b}^{*} \right\}}} + {\left( {\underset{2 \times n_{h}^{*}}{\left\lbrack Y_{h}^{*} \right\rbrack} - {\underset{2 \times n}{\left\lbrack Y_{e}^{*} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e}^{*} \right\rbrack}}^{- 1}\underset{n \times n_{h}^{*}}{\left\lbrack K_{h}^{*} \right\rbrack}}} \right)\underset{n_{h}^{*} \times 1}{\left\{ s_{h}^{*} \right\}}}}} & (361) \end{matrix}$

The first term is a function having the known part s_(b)*, and becomes a displacement {u_(B)*} that satisfies the inhomogeneous adjoint boundary condition. The second term is a function having the unknown part s_(h)*, and becomes a displacement {u_(H)*} that satisfies the homogeneous adjoint boundary condition. In other words, we define:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 195} \right\rbrack & \; \\ {\underset{2 \times n_{b}^{*}}{\left\lbrack \chi_{B}^{*} \right\rbrack} \equiv {\underset{2 \times n_{b}^{*}}{\left\lbrack Y_{b}^{*} \right\rbrack} - {\underset{2 \times n}{\left\lbrack Y_{e}^{*} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e}^{*} \right\rbrack}}^{- 1}\underset{n \times n_{b}^{*}}{\left\lbrack K_{b}^{*} \right\rbrack}}}} & (362) \\ {\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack} \equiv {\underset{2 \times n_{h}^{*}}{\left\lbrack Y_{h}^{*} \right\rbrack} - {\underset{2 \times n}{\left\lbrack Y_{e}^{*} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e}^{*} \right\rbrack}}^{- 1}\underset{n \times n_{h}^{*}}{\left\lbrack K_{h}^{*} \right\rbrack}}}} & (363) \end{matrix}$

Then, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 196} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u_{B}^{*} \right\}} \equiv {\underset{2 \times n_{b}^{*}}{\left\lbrack \chi_{B}^{*} \right\rbrack}\underset{n_{b}^{*} \times 1}{\left\{ s_{b}^{*} \right\}}}} & (364) \\ {\underset{2 \times 1}{\left\{ u_{H}^{*} \right\}} \equiv {\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ s_{h}^{*} \right\}}}} & (365) \end{matrix}$

And we find:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 197} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u^{*} \right\}} = {\underset{2 \times 1}{\left\{ u_{B}^{*} \right\}} + \underset{2 \times 1}{\left\{ u_{H}^{*} \right\}}}} & (366) \end{matrix}$

This equation shows details of eq. (191), and we find that the displacement {u*} is expressed with the displacement {u_(B)*} that satisfies the inhomogeneous adjoint boundary condition and the displacement {u_(H)} that satisfies the homogeneous adjoint boundary condition.

According to eqs. (333), (356), and (362), the matrix [χ_(B)*] is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 198} \right\rbrack & \; \\ {\underset{2 \times n_{b}^{*}}{\left\lbrack \chi_{B}^{*} \right\rbrack} \equiv {{\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{b}^{*}}{\left\lbrack c_{b}^{*} \right\rbrack}} - {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e}^{*} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e}^{*} \right\rbrack}}^{- 1}\underset{n \times n_{b}^{*}}{\left\lbrack K_{b}^{*} \right\rbrack}}}} & (367) \end{matrix}$

According to eq. (293), we find that the first term of this equation is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 199} \right\rbrack & \; \\ {{\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times n_{b}^{*}}{\left\lbrack c_{b}^{*} \right\rbrack}} = {{\begin{bmatrix} \underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}\begin{bmatrix} \underset{n \times n_{b}^{*}}{\left\lbrack c_{b}^{*} \right\rbrack} \\ \underset{\lambda \times n_{b}^{*}}{\lbrack 0\rbrack} \end{bmatrix}} = {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\begin{bmatrix} \underset{n \times n_{b}^{*}}{\left\lbrack c_{b}^{*} \right\rbrack} \\ \underset{\lambda \times n_{b}^{*}}{\lbrack 0\rbrack} \end{bmatrix}}}} & (368) \end{matrix}$

Focusing this, we can give the following:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 200} \right\rbrack & \; \\ {\underset{{({n + \lambda})} \times n_{b}^{*}}{\left\lbrack H_{b}^{*} \right\rbrack} \equiv {\begin{bmatrix} \underset{n \times n_{b}^{*}}{\left\lbrack c_{b}^{*} \right\rbrack} \\ \underset{\lambda \times n_{b}^{*}}{\lbrack 0\rbrack} \end{bmatrix} - {\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e}^{*} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e}^{*} \right\rbrack}}^{- 1}\underset{n \times n_{b}^{*}}{\left\lbrack K_{b}^{*} \right\rbrack}}}} & (369) \end{matrix}$

Then, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 201} \right\rbrack & \; \\ {\underset{2 \times n_{b}^{*}}{\left\{ \chi_{B}^{*} \right\}} \equiv {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n_{b}^{*}}{\left\lbrack H_{b}^{*} \right\rbrack}}} & (370) \end{matrix}$

According to eqs. (356), (357), and (363), the matrix [ψ_(h)*] is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 202} \right\rbrack & \; \\ {\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack} \equiv {{\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack c_{h}^{*} \right\rbrack}} - {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e}^{*} \right\rbrack}{\underset{n \times n}{\left\lbrack K_{e}^{*} \right\rbrack}}^{- 1}\underset{n \times n_{h}^{*}}{\left\lbrack K_{h}^{*} \right\rbrack}}}} & (371) \end{matrix}$

Assuming

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 203} \right\rbrack & \; \\ {{\underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack H_{h}^{*} \right\rbrack} \equiv {\underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack c_{h}^{*} \right\rbrack} - {\underset{{({n + \lambda})} \times n}{\left\lbrack c_{e}^{*} \right\rbrack}{\underset{n \times m}{\left\lbrack K_{e}^{*} \right\rbrack}}^{- 1}\underset{n \times n_{h}^{*}}{\left\lbrack K_{h}^{*} \right\rbrack}}}},} & (372) \end{matrix}$

then, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 204} \right\rbrack & \; \\ {\underset{2 \times n_{h}^{*}}{\left\{ \psi_{h}^{*} \right\}} \equiv {\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack H_{h}^{*} \right\rbrack}}} & (373) \end{matrix}$

According to eqs. (370) and (373), we find that the matrices [χ_(B)*] and [ψ_(h)*] are separated into a function part [Γ] and coefficient parts [H_(b)*] and [H_(h)*]. Further, the displacement {u*} is expressed as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 205} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u^{*} \right\}} = {{\underset{2 \times 1}{\left\{ u_{B}^{*} \right\}} + \underset{2 \times 1}{\left\{ u_{H}^{*} \right\}}} = {{\underset{2 \times n_{b}^{*}}{\left\lbrack _{B}^{*} \right\rbrack}\underset{n_{b}^{*} \times 1}{\left\{ s_{b}^{*} \right\}}} + {\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ s_{h}^{*} \right\}}}}}} & (374) \end{matrix}$

From this, we find that an arbitrary nodal known part s_(b)* composes a function that satisfies the inhomogeneous adjoint boundary condition, and an arbitrary unknown part s_(h)* composes a function that satisfies the homogeneous adjoint boundary condition. That the displacement {u_(H)*} satisfies the homogeneous adjoint boundary condition with respect to the unknown part s_(h)* is achieved by the effects of the matrix [ψ_(h)*]. Therefore, let the coefficient vector be {e_(h)*}, and we can give the dual eigenfunction {φ*} as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 206} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ \varphi^{*} \right\}} \equiv {\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ e_{h}^{*} \right\}}}} & (375) \end{matrix}$

At the same time, we can give the variation {δφ*} as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 207} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ {\delta\varphi}^{*} \right\}} \equiv {\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ {\delta \; e_{h}^{*}} \right\}}}} & (376) \end{matrix}$

The matrix [ψ_(h)*], which plays an important role, is referred to as a “dual trial function”.

8.14 Relationship of Numbers

According to eqs. (259), (320), and (252), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 208} \right\rbrack & \; \\ \begin{matrix} {{n_{b} + n_{b}^{*}} = {n_{U_{b}} + n_{F_{b}} + n_{U_{b}^{*}} + n_{F_{b}^{*}}}} \\ {= {{n_{U_{b}} + n_{F_{b}} + n_{F_{v}} + n_{U_{v}}} = {2n}}} \end{matrix} & (377) \end{matrix}$

According to eqs. (278), (339), and (377), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 209} \right\rbrack & \; \\ \begin{matrix} {{n_{h} + n_{h}^{*}} = {\left( {\lambda + n - n_{b}} \right) + \left( {\lambda + n - n_{b}^{*}} \right)}} \\ {= {{{2\lambda} + {2n} - \left( {n_{b} + n_{b}^{*}} \right)} = {2\lambda}}} \end{matrix} & (378) \end{matrix}$

8.15 Determination of Primal Eigenfunction and Dual Eigenfunction

The primal simultaneous differential equation (105) and the dual simultaneous differential equation (106) are expressed in matrix as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 210} \right\rbrack & \; \\ {{\sum\limits_{j}\; {L_{ij}\varphi_{j}}} = {\left. {\lambda \; w_{i}\varphi_{i}^{*}}\Rightarrow{\begin{bmatrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{bmatrix}\begin{Bmatrix} \varphi_{1} \\ \varphi_{2} \end{Bmatrix}} \right. = {{\lambda \begin{bmatrix} w_{1} & 0 \\ 0 & w_{2} \end{bmatrix}}\begin{Bmatrix} \varphi_{1}^{*} \\ \varphi_{2}^{*} \end{Bmatrix}}}} & (379) \\ {{\sum\limits_{j}\; {L_{ji}^{*}\varphi_{j}^{*}}} = {\left. {\lambda \; w_{i}\varphi_{i}}\Rightarrow{\begin{bmatrix} L_{11}^{*} & L_{21}^{*} \\ L_{12}^{*} & L_{22}^{*} \end{bmatrix}\begin{Bmatrix} \varphi_{1}^{*} \\ \varphi_{2}^{*} \end{Bmatrix}} \right. = {{\lambda \begin{bmatrix} w_{1} & 0 \\ 0 & w_{2} \end{bmatrix}}\begin{Bmatrix} \varphi_{1} \\ \varphi_{2} \end{Bmatrix}}}} & (380) \end{matrix}$

Let differential operators be:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 211} \right\rbrack & \; \\ {\underset{2 \times 2}{\lbrack L\rbrack} = \begin{bmatrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{bmatrix}} & (381) \\ {\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack} = \begin{bmatrix} L_{11}^{*} & L_{21}^{*} \\ L_{12}^{*} & L_{22}^{*} \end{bmatrix}} & (382) \end{matrix}$

And let a weight be:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 212} \right\rbrack & \; \\ {\underset{2 \times 2}{\lbrack w\rbrack} = \begin{bmatrix} w_{1} & 0 \\ 0 & w_{2} \end{bmatrix}} & (383) \end{matrix}$

Then, the primal and dual simultaneous differential equations are transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 213} \right\rbrack & \; \\ {{\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times 1}{\left\{ \varphi \right\}}} = {\lambda \underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times 1}{\left\{ \varphi^{*} \right\}}}} & (384) \\ {{\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times 1}{\left\{ \varphi^{*} \right\}}} = {\lambda \underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times 1}{\left\{ \varphi \right\}}}} & (385) \end{matrix}$

According to the direct variational method of eq. (122), multiplying eqs. (384) and (385) by variations {δφ*} and {δφ}, respectively, and integrating the same, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 214} \right\rbrack & \; \\ {{\int_{S}{{\underset{1 \times 2}{\left\{ {\delta\varphi}^{*} \right\}}}^{T}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times 1}{\left\{ \varphi \right\}}\ {s}}} = {\lambda {\int_{S}{{\underset{1 \times 2}{\left\{ {\delta\varphi}^{*} \right\}}}^{T}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times 1}{\left\{ \varphi^{*} \right\}}\ {s}}}}} & (386) \\ {{\int_{S}{{\underset{1 \times 2}{\left\{ {\delta\varphi} \right\}}}^{T}\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times 1}{\left\{ \varphi^{*} \right\}}\ {s}}} = {\lambda {\int_{S}{\underset{1 \times 2}{\left\{ {\delta\varphi} \right\}^{T}}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times 1}{\left\{ \varphi \right\}}\ {s}}}}} & (387) \end{matrix}$

Substituting the eigenfunctions {φ} and {φ*} of eqs. (315) and (375), and the variations {δφ} and {δφ*} of eqs. (316) and (376) into the foregoing equations, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 215} \right\rbrack & \; \\ {{\underset{1 \times n_{h}^{*}}{\left\{ {\delta \; e_{h}^{*}} \right\}^{T}}{\int_{S}{\underset{n_{h}^{*} \times 2}{\left\lbrack \psi_{h}^{*} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\ {s}\underset{n_{h} \times 1}{\left\{ e_{h} \right\}}}}} = {\lambda \underset{1 \times n_{h}^{*}}{\left\{ {\delta \; e_{h}^{*}} \right\}^{T}}{\int_{S}{\underset{n_{h}^{*} \times 2}{\left\lbrack \psi_{h}^{*} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack \ }{s}\underset{n_{h}^{*} \times 1}{\left\{ e_{h}^{*} \right\}}}}}} & (388) \\ {{\underset{1 \times n_{h}}{\left\{ {\delta \; e_{h}} \right\}^{T}}{\int_{S}{\underset{n_{h} \times 2}{\left\lbrack \psi_{h} \right\rbrack^{T}}\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack \ }{s}\underset{n_{h}^{*} \times 1}{\left\{ e_{h}^{*} \right\}}}}} = {\lambda \underset{1 \times n_{h}}{\left\{ {\delta \; e_{h}} \right\}^{T}}{\int_{S}{\underset{n_{h} \times 2}{\left\lbrack \psi_{h} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\ {s}\underset{n_{h} \times 1}{\left\{ e_{h} \right\}}}}}} & (389) \end{matrix}$

Let the integration results on the left sides of the equations be:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 216} \right\rbrack & \; \\ {\underset{n_{h}^{*} \times n_{h}}{\left\lbrack A_{h} \right\rbrack} \equiv {\int_{S}{\underset{n_{h}^{*} \times 2}{\left\lbrack \psi_{h}^{*} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack \ }{s}}}} & (390) \\ {\underset{n_{h} \times n_{h}^{*}}{\left\lbrack A_{h}^{*} \right\rbrack} \equiv {\int_{S}{\underset{n_{h} \times 2}{\left\lbrack \psi_{h} \right\rbrack^{T}}\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack}\ {s}}}} & (391) \end{matrix}$

Let the integrations results on the right sides of the equations be:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 217} \right\rbrack & \; \\ {\underset{n_{h} \times n_{h}}{\left\lbrack B_{h} \right\rbrack} \equiv {\int_{S}{\underset{n_{h} \times 2}{\left\lbrack \psi_{h} \right\rbrack^{T}}\underset{2 \times}{\lbrack w\rbrack}\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\ {s}}}} & (392) \\ {\underset{n_{h}^{*} \times n_{h}^{*}}{\left\lbrack B_{h}^{*} \right\rbrack} \equiv {\int_{S}{\underset{n_{h}^{*} \times 2}{\left\lbrack \psi_{h}^{*} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack \ }{s}}}} & (393) \end{matrix}$

[B_(h)] and [B_(h)*] are both symmetric matrices. Further, the primal trial function [ψ_(h)] satisfies the homogeneous boundary condition, and the dual trial function [ψ_(h)*] satisfies the homogeneous adjoint boundary condition. Therefore, partial integration of the same gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 218} \right\rbrack & \; \\ {{\underset{n_{h}^{*} \times n_{h}}{\left\lbrack A_{h} \right\rbrack} \equiv {\int_{S}{\underset{n_{h}^{*} \times 2}{\left\lbrack \psi_{h}^{*} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\ {s}}}} = {\left( {\int_{S}{\underset{n_{h} \times 2}{\left\lbrack \psi_{h} \right\rbrack^{T}}\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack}\ {s}}} \right)^{T} = \underset{n_{h}^{*} \times n_{h}}{\left\lbrack A_{h}^{*} \right\rbrack^{T}}}} & (394) \end{matrix}$

In other words, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 219} \right\rbrack & \; \\ {\underset{n_{h} \times n_{h}^{*}}{\left\lbrack A_{h}^{*} \right\rbrack} = \underset{n_{h} \times n_{h}^{*}}{\left\lbrack A_{h} \right\rbrack^{T}}} & (395) \end{matrix}$

With use of eqs. (390) to (393), eqs. (388) and (389) are transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 220} \right\rbrack & \; \\ {{{\underset{1 \times n_{h}^{*}}{\left\{ {\delta \; e_{h}^{*}} \right\}}}^{T}\underset{n_{h}^{*} \times n_{h}}{\left\lbrack A_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ e_{h} \right\}}} = {\lambda {\underset{1 \times n_{h}^{*}}{\left\{ {\delta \; e_{h}^{*}} \right\}}}^{T}\underset{n_{h}^{*} \times n_{h}^{*}}{\left\lbrack B_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ e_{h}^{*} \right\}}}} & (396) \\ {{{\underset{1 \times n_{h}}{\left\{ {\delta \; e_{h}} \right\}}}^{T}\underset{n_{h} \times n_{h}^{*}}{\left\lbrack A_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ e_{h}^{*} \right\}}} = {\lambda {\underset{1 \times n_{h}}{\left\{ {\delta \; e_{h}} \right\}}}^{T}\underset{n_{h} \times n_{h}}{\left\lbrack B_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ e_{h} \right\}}}} & (397) \end{matrix}$

In order that these are established with respect to an arbitrary variation, the following has to be established:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 221} \right\rbrack & \; \\ {{\underset{n_{h}^{*} \times n_{h}}{\left\lbrack A_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ e_{h} \right\}}} = {\lambda \underset{n_{h}^{*} \times n_{h}^{*}}{\left\lbrack B_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ e_{h}^{*} \right\}}}} & (398) \\ {{\underset{n_{h} \times n_{h}^{*}}{\left\lbrack A_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ e_{h}^{*} \right\}}} = {\lambda \underset{n_{h} \times n_{h}}{\left\lbrack B_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ e_{h} \right\}}}} & (399) \end{matrix}$

Mutual substitution of eq. (398) and eq. (399) with each other gives the following equations on each eigenvector:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 222} \right\rbrack & \; \\ {{\underset{n_{h} \times n_{h}^{*}}{\left\lbrack A_{h}^{*} \right\rbrack}{\underset{n_{h} \times n_{h}}{\left\lbrack B_{h} \right\rbrack}}^{- 1}\underset{n_{h}^{*} \times n_{h}}{\left\lbrack A_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ e_{h} \right\}}} = {\lambda^{2}\underset{n_{h} \times n_{h}}{\left\lbrack B_{h} \right\rbrack}\underset{n_{h} \times 1}{\left\{ e_{h} \right\}}}} & (400) \\ {{\underset{n_{h}^{*} \times n_{h}}{\left\lbrack A_{h} \right\rbrack}\underset{n_{h} \times n_{h}}{\left\lbrack B_{h} \right\rbrack^{- 1}}\underset{n_{h} \times n_{h}^{*}}{\left\lbrack A_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ e_{h}^{*} \right\}}} = {\lambda^{2}\underset{n_{h}^{*} \times n_{h}^{*}}{\left\lbrack B_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times 1}{\left\{ e_{h}^{*} \right\}}}} & (401) \end{matrix}$

By solving these, we obtain an eigenvalue λ and eigenvectors {e_(h)}, {e_(h)*}. By contraries, combining eqs. (398) and (399) into one, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 223} \right\rbrack & \; \\ {{\underset{2\; \lambda \times 2\; \lambda}{\left\lbrack A_{s} \right\rbrack}\underset{2\; \lambda \times 1}{\left\{ e_{s} \right\}}} = {\lambda \underset{\mspace{2mu} {2\; \lambda \times 2\; \lambda}}{\left\lbrack B_{s} \right\rbrack}\underset{2\; \lambda \times 1}{\left\{ e_{s} \right\}}}} & (402) \\ \left\lbrack {{Formula}\mspace{14mu} 224} \right\rbrack & \; \\ {{\underset{2\; \lambda \times 2\; \lambda}{\left\lbrack A_{s} \right\rbrack} \equiv \begin{bmatrix} \underset{n_{h}^{*} \times n_{h}^{*}}{\lbrack 0\rbrack} & \underset{n_{h}^{*} \times n_{h}}{\left\lbrack A_{h} \right\rbrack} \\ \underset{n_{h} \times n_{h}^{*}}{\left\lbrack A_{h}^{*} \right\rbrack} & \underset{n_{h} \times n_{h}}{\lbrack 0\rbrack} \end{bmatrix}} = \begin{bmatrix} \underset{n_{h}^{*} \times n_{h}^{*}}{\lbrack 0\rbrack} & \underset{n_{h}^{*} \times n_{h}}{\left\lbrack A_{h} \right\rbrack} \\ \underset{n_{h} \times n_{h}^{*}}{\left\lbrack A_{h} \right\rbrack^{T}} & \underset{n_{h} \times n_{h}}{\lbrack 0\rbrack} \end{bmatrix}} & (403) \\ {\underset{\mspace{2mu} {2\; \lambda \times 2\; \lambda}}{\left\lbrack B_{s} \right\rbrack} \equiv \begin{bmatrix} \underset{n_{h}^{*} \times n_{h}^{*}}{\left\lbrack B_{h}^{*} \right\rbrack} & \underset{n_{h}^{*} \times n_{h}}{\lbrack 0\rbrack} \\ \underset{n_{h} \times n_{h}^{*}}{\lbrack 0\rbrack} & \underset{n_{h} \times n_{h}}{\left\lbrack B_{h} \right\rbrack} \end{bmatrix}} & (404) \\ {\underset{2\; \lambda \times 1}{\left\{ e_{s} \right\}} \equiv \begin{Bmatrix} \underset{n_{h}^{*} \times 1}{\left\{ e_{h}^{*} \right\}} \\ \underset{n_{h} \times 1}{\left\{ e_{h} \right\}} \end{Bmatrix}} & (405) \end{matrix}$

It should be noted that the equal sign in eq. (403) results from eq. (395).

By solving this, we obtain the eigenvalue A and the eigenvectors {e_(h)} and {e_(h)*}. Since [A_(s)] and [B_(s)] are both symmetric matrices, the eigenvalue λ is a real number, and the eigenvectors {e_(s)} are orthogonal each other, when [A_(s)] or [B_(s)] is interposed between them.

The calculation by eqs. (400) and (401) results in that the eigenvalue problem has a small size. However, the combination of the eigenvectors {e_(h)} and {e_(h)*} has to be dealt with carefully. On the other hand, the calculation by eq. (402) results in that the eigenvalue problem has a large size, but the combination of the eigenvectors {e_(h)} and {e_(h)*} can be obtained simultaneously, which makes it unnecessary to confirm the combination. Which method should be used may be decided according to the memory size, the calculation speed, for example. As eq. (402) is easier to handle from the viewpoint of formulation, the following description is based on this.

From eq. (402), 2 l combinations of the eigenvalue A and the eigenvector {e_(s)} are obtained. The number of positive eigenvalues and the number of negative eigenvalues coincide. Let the number be m_(p). This coincidence can be derived from the fact that deformation patterns of eqs. (398) and (399) in which λ and {e_(h)*} are sign-inverted while {e_(h)} remains without inversion of the sign, also satisfy the simultaneous equations. This is described also in Section 4.4. In the case where [A_(s)] has null space, the eigenvalue is zero. Let the number of null space of [A_(h)] be m₀.

By returning components {e_(h)} and {e_(h)*} of the eigenvector {e_(s)} belonging to the eigenvalue λ to eqs. (315) and (375), we obtain eigenfunctions {φ} and {φ*}.

Arranging m_(p) eigenfunctions {φ}, {φ*} and m_(p) eigenvectors {e_(h)}, {e_(h)*} belonging to a positive eigenvalue in the column direction (horizontal direction) and expressing them as [φ_(p)], [φ_(p)*] and [e_(p)], [e_(p)*], we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 225} \right\rbrack & \; \\ {\underset{2 \times m_{p}}{\left\lbrack \varphi_{p} \right\rbrack} \equiv {\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\underset{n_{h} \times m_{p}}{\left\lbrack e_{p} \right\rbrack}}} & (406) \\ {\underset{2 \times m_{p}}{\left\lbrack \varphi_{p}^{*} \right\rbrack} \equiv {\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack}\underset{n_{h}^{*} \times m_{p}}{\left\lbrack e_{p}^{*} \right\rbrack}}} & (407) \end{matrix}$

Arranging m_(p) eigenvalues on the diagonal, we define:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 226} \right\rbrack & \; \\ {\underset{m_{p} \times m_{p}}{\left\lbrack \lambda_{p} \right\rbrack} \equiv \begin{bmatrix} \lambda_{1} & 0 & \Lambda & 0 \\ 0 & \lambda_{2} & \Lambda & 0 \\ M & M & O & M \\ 0 & 0 & \Lambda & \lambda_{m} \end{bmatrix}} & (408) \end{matrix}$

Then, we obtain primal and dual simultaneous differential equations as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 227} \right\rbrack & \; \\ {{\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times m_{p}}{\left\lbrack \varphi_{p} \right\rbrack}} = {\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times m_{p}}{\left\lbrack \varphi_{p}^{*} \right\rbrack}\underset{m_{p} \times m_{p}}{\left\lbrack \lambda_{p} \right\rbrack}}} & (409) \\ {{\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times m_{p}}{\left\lbrack \varphi_{p}^{*} \right\rbrack}} = {\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times m_{p}}{\left\lbrack \varphi_{p} \right\rbrack}\underset{m_{p} \times m_{p}}{\left\lbrack \lambda_{p} \right\rbrack}}} & (410) \end{matrix}$

The normalization with the orthogonality of the eigenfunctions allows [e_(p)], [e_(p)*] to satisfy:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 228} \right\rbrack & \; \\ {{\int_{S}{\underset{m_{p} \times 2}{\left\lbrack \varphi_{p} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times m_{p}}{\left\lbrack \varphi_{p} \right\rbrack}\ {s}}} = \underset{m_{p} \times m_{p}}{\lbrack I\rbrack}} & (411) \\ {{\int_{S}{\underset{m_{p} \times 2}{\left\lbrack \varphi_{p}^{*} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times m_{p}}{\left\lbrack \varphi_{p}^{*} \right\rbrack}\ {s}}} = \underset{m_{p} \times m_{p}}{\lbrack I\rbrack}} & (412) \end{matrix}$

As a result, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 229} \right\rbrack & \; \\ {{\int_{S}{{\underset{m_{p} \times 2}{\left\lbrack \varphi_{p}^{*} \right\rbrack}}^{T}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times m_{p}}{\left\lbrack \varphi_{p} \right\rbrack}\ {s}}} = \underset{m_{p} \times m_{p}}{\left\lbrack \lambda_{p} \right\rbrack}} & (413) \\ {{\int_{S}{{\underset{m_{p} \times 2}{\left\lbrack \varphi_{p} \right\rbrack}}^{T}\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times m_{p}}{\left\lbrack \varphi_{p}^{*} \right\rbrack}\ {s}}} = \underset{m_{p} \times m_{p}}{\left\lbrack \lambda_{p} \right\rbrack}} & (414) \end{matrix}$

m₀ eigenfunctions {φ} and eigenvectors {e_(h)} belonging to a zero eigenvalue are arranged in the column direction (horizontal direction). Let them be [φ₀] and [e₀], respectively. Then, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 230} \right\rbrack & \; \\ {\underset{2 \times m_{o}}{\left\lbrack \varphi_{o} \right\rbrack} \equiv {\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\underset{n_{h} \times m_{o}}{\left\lbrack e_{o} \right\rbrack}}} & (415) \end{matrix}$

According to the orthogonality of the eigenfunctions, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 231} \right\rbrack & \; \\ {{\int_{S}{\underset{m_{p} \times 2}{\left\lbrack \varphi_{p}^{*} \right\rbrack}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times m_{o}}{\left\lbrack \varphi_{o} \right\rbrack}\ {s}}} = \underset{m_{p} \times m_{o}}{\lbrack 0\rbrack}} & (416) \end{matrix}$

8.16 Details of Matrices

By transforming [A_(h)] of eq. (390) according to eqs. (293), (313), and (373), we obtain:

$\begin{matrix} {\; \left\lbrack {{Formula}\mspace{14mu} 232} \right\rbrack} & \; \\ \begin{matrix} {\underset{n_{h}^{*} \times n_{h}}{\left\lbrack A_{h} \right\rbrack} \equiv {\int_{S}{{\underset{{n_{h}^{*} \times 2}\;}{\left\lbrack \psi_{h}^{*} \right\rbrack}}^{T}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\ {s}}}} \\ {= {\underset{n_{h}^{*} \times {({n + \lambda})}}{\left\lbrack H_{h}^{*} \right\rbrack^{T}}{\int_{S}{\underset{{({n + \lambda})} \times 2}{\lbrack\Gamma\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\ {s}\underset{{({n + \lambda})} \times n_{h}}{\left\lbrack H_{h} \right\rbrack}}}}} \\ {= {\underset{n_{h}^{*} \times {({n + \lambda})}}{\left\lbrack H_{h}^{*} \right\rbrack^{T}}{\int_{S}{\begin{bmatrix} {\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T} \\ \underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}} \end{bmatrix}{\underset{2 \times 2}{\lbrack L\rbrack}\begin{bmatrix} \underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}}\ {s}\; \underset{{({n + \lambda})} \times n_{h}}{\left\lbrack H_{h} \right\rbrack}}}}} \\ {= {\underset{n_{h}^{*} \times {({n + \lambda})}}{\left\lbrack H_{h}^{*} \right\rbrack^{T}}{\int_{S}{\left\lbrack {\begin{matrix} {\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T} \\ \underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}} \end{matrix}\begin{matrix} {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \\ {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}{\underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack}}^{T}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \end{matrix}} \right\rbrack {s}\; \underset{{({n + \lambda})} \times n_{h}}{\left\lbrack H_{h} \right\rbrack}}}}} \end{matrix} & (417) \end{matrix}$

By transforming [A_(h)*] of eq. (391) according to eqs. (293), (313), and (373), we obtain:

$\begin{matrix} {\; \left\lbrack {{Formula}\mspace{14mu} 233} \right\rbrack} & \; \\ \begin{matrix} {\underset{n_{h} \times n_{h}^{*}}{\left\lbrack A_{h}^{*} \right\rbrack} \equiv {\int_{S}{{\underset{{n_{h} \times 2}\;}{\left\lbrack \psi_{h} \right\rbrack}}^{T}\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack}\ {s}}}} \\ {= {\underset{n_{h} \times {({n + \lambda})}}{\left\lbrack H_{h} \right\rbrack^{T}}{\int_{S}{\underset{{({n + \lambda})} \times 2}{\lbrack\Gamma\rbrack^{T}}\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\ {s}\underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack H_{h}^{*} \right\rbrack}}}}} \\ {= {\underset{n_{h} \times {({n + \lambda})}}{\left\lbrack H_{h} \right\rbrack^{T}}{\int_{S}{\begin{bmatrix} {\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T} \\ \underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}} \end{bmatrix}{\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\begin{bmatrix} \underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}}\ {s}\; \underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack H_{h}^{*} \right\rbrack}}}}} \\ {= {\underset{n_{h} \times {({n + \lambda})}}{\left\lbrack H_{h} \right\rbrack^{T}}{\int_{S}{\left\lbrack {\begin{matrix} {\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T} \\ \underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}} \end{matrix}\begin{matrix} {\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T}\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \\ {\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}{\underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack}}^{T}\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \end{matrix}} \right\rbrack {s}\; \underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack H_{h}^{*} \right\rbrack}}}}} \end{matrix} & (418) \end{matrix}$

By transforming [B_(h)] of eq. (392) according to eqs. (293) and (313), we obtain:

$\begin{matrix} {\; \left\lbrack {{Formula}\mspace{14mu} 234} \right\rbrack} & \; \\ \begin{matrix} {\underset{n_{h} \times n_{h}}{\left\lbrack B_{h} \right\rbrack} \equiv {\int_{S}{{\underset{{n_{h} \times 2}\;}{\left\lbrack \psi_{h} \right\rbrack}}^{T}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\ {s}}}} \\ {= {\underset{n_{h} \times {({n + \lambda})}}{\left\lbrack H_{h} \right\rbrack^{T}}{\int_{S}{\underset{{({n + \lambda})} \times 2}{\lbrack\Gamma\rbrack^{T}}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\ {s}\underset{{({n + \lambda})} \times n_{h}}{\left\lbrack H_{h} \right\rbrack}}}}} \\ {= {\underset{n_{h} \times {({n + \lambda})}}{\left\lbrack H_{h} \right\rbrack^{T}}{\int_{S}{\begin{bmatrix} {\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T} \\ \underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}} \end{bmatrix}{\underset{2 \times 2}{\lbrack w\rbrack}\begin{bmatrix} \underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}}\ {s}\; \underset{{({n + \lambda})} \times n_{h}}{\left\lbrack H_{h} \right\rbrack}}}}} \\ {= {\underset{n_{h} \times {({n + \lambda})}}{\left\lbrack H_{h} \right\rbrack^{T}}{\int_{S}{\left\lbrack {\begin{matrix} {\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T} \\ \underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}} \end{matrix}\begin{matrix} {\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \\ {\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}{\underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack}}^{T}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \end{matrix}} \right\rbrack {s}\; \underset{{({n + \lambda})} \times n_{h}}{\left\lbrack H_{h} \right\rbrack}}}}} \end{matrix} & (419) \end{matrix}$

This clearly shows that [B_(h)] becomes a symmetric matrix. By transforming [B_(h)*] of eq. (393) according to eqs. (293) and (373), we obtain:

$\begin{matrix} {\; \left\lbrack {{Formula}\mspace{14mu} 235} \right\rbrack} & \; \\ \begin{matrix} {\underset{n_{h}^{*} \times n_{h}^{*}}{\left\lbrack B_{h}^{*} \right\rbrack} \equiv {\int_{S}{{\underset{{n_{h}^{*} \times 2}\;}{\left\lbrack \psi_{h}^{*} \right\rbrack}}^{T}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times n_{h}^{*}}{\left\lbrack \psi_{h}^{*} \right\rbrack}\ {s}}}} \\ {= {\underset{n_{h}^{*} \times {({n + \lambda})}}{\left\lbrack H_{h}^{*} \right\rbrack^{T}}{\int_{S}{\underset{{({n + \lambda})} \times 2}{\lbrack\Gamma\rbrack^{T}}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}\ {s}\underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack H_{h}^{*} \right\rbrack}}}}} \\ {= {\underset{n_{h}^{*} \times {({n + \lambda})}}{\left\lbrack H_{h}^{*} \right\rbrack^{T}}{\int_{S}{\begin{bmatrix} {\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T} \\ \underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}} \end{bmatrix}{\underset{2 \times 2}{\lbrack w\rbrack}\begin{bmatrix} \underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}}\ {s}\; \underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack H_{h}^{*} \right\rbrack}}}}} \\ {= {\underset{n_{h}^{*} \times {({n + \lambda})}}{\left\lbrack H_{h}^{*} \right\rbrack^{T}}{\int_{S}{\left\lbrack {\begin{matrix} {\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T} \\ \underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}} \end{matrix}\begin{matrix} {\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack}}^{T}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \\ {\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}{\underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack}}^{T}\underset{2 \times 2}{\lbrack w\rbrack}\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \end{matrix}} \right\rbrack {s}\; \underset{{({n + \lambda})} \times n_{h}^{*}}{\left\lbrack H_{h}^{*} \right\rbrack}}}}} \end{matrix} & (420) \end{matrix}$

This clearly shows that [B_(h)*] becomes a symmetric matrix.

8.17 Eigenfunction Method

Expressing the simultaneous partial differential equation (23) of the primal problem in matrix, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 236} \right\rbrack & \; \\ {{\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times 1}{\left\{ u \right\}}} = \underset{2 \times 1}{\left\{ f \right\}}} & (421) \end{matrix}$

As the displacement {u} can be expressed by eq. (314), this equation is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 237} \right\rbrack & \; \\ {{\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times 1}{\left\{ u_{H} \right\}}} = {\underset{2 \times 1}{\left\{ f \right\}} - {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times 1}{\left\{ u_{B} \right\}}}}} & (422) \end{matrix}$

On the other hand, the displacement {u_(H)} is expressed as follows with the eigenfunctions [φ_(p)] and [φ₀], with the coefficient vectors being defined to be {a_(p)}, {a₀}, and with use of eqs. (406) and (415):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 238} \right\rbrack & \; \\ {{\underset{2 \times 1}{\left\{ u_{H} \right\}} \equiv {{\underset{2 \times m_{p}}{\left\lbrack \varphi_{p} \right\rbrack}\underset{m_{p} \times 1}{\left\{ a_{p} \right\}}} + {\underset{2 \times m_{o\;}}{\left\lbrack \varphi_{o} \right\rbrack}\underset{m_{o} \times 1}{\left\{ a_{o} \right\}}}}} = {\underset{2 \times n_{h}}{\left\lbrack \psi_{h} \right\rbrack}\left( {{\underset{n_{h} \times m_{p}}{\left\lbrack e_{p} \right\rbrack}\underset{m_{p} \times 1}{\left\{ a_{p} \right\}}} + {\underset{n_{h} \times m_{o}}{\left\lbrack e_{o} \right\rbrack}\underset{m_{o} \times 1}{\left\{ a_{o} \right\}}}} \right)}} & (423) \end{matrix}$

Comparing this equation and eq. (305), we find that the unknown part s_(h) is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 239} \right\rbrack & \; \\ {\underset{n_{h} \times 1}{\left\{ s_{h} \right\}} = {{\underset{n_{h} \times m_{p}}{\left\lbrack e_{p} \right\rbrack}\underset{m_{p} \times 1}{\left\{ a_{p} \right\}}} + {\underset{n_{h} \times m_{o}}{\left\lbrack e_{o} \right\rbrack}\underset{m_{o} \times 1}{\left\{ a_{o} \right\}}}}} & (424) \end{matrix}$

In other words, this indicates that when the coefficient vectors {a_(p)} and {a₀} are settled, the unknown part s_(h) is determined. According to eqs. (422) and (423), the simultaneous partial differential equation is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 240} \right\rbrack & \; \\ {{{\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times m_{p}}{\left\lbrack \varphi_{p} \right\rbrack}\underset{m_{p} \times 1}{\left\{ a_{p} \right\}}} + {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times m_{o}}{\left\lbrack \varphi_{o} \right\rbrack}\underset{m_{o} \times 1}{\left\{ a_{o} \right\}}}} = {\underset{2 \times 1}{\left\{ f \right\}} - {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times 1}{\left\{ u_{B} \right\}}}}} & (425) \end{matrix}$

When both sides are multiplied by [φ_(p)*]^(T) so that an inner product is taken, the second term of the left side of the equation becomes zero according to eq. (416), and we obtain:

$\begin{matrix} {\mspace{20mu} \left\lbrack {{Formula}\mspace{14mu} 241} \right\rbrack} & \; \\ {{\int_{S}{\underset{m_{p} \times 2}{\left\lbrack \varphi_{p}^{*} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times m_{p}}{\left\lbrack \varphi_{p} \right\rbrack}{s}\underset{m_{p} \times 1}{\left\{ a_{p} \right\}}}} = {{\int_{S}{\underset{m_{p} \times 2}{\left\lbrack \varphi_{p}^{*} \right\rbrack^{T}}\underset{2 \times 1}{\left\{ f \right\}}{s}}} - {\int_{S}{\underset{m_{p} \times 2}{\left\lbrack \varphi_{p}^{*} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times 1}{\left\{ u_{B} \right\}}{s}}}}} & (426) \end{matrix}$

According to eqs. (304), (407), and (413), this equation is transformed to:

$\begin{matrix} {\mspace{20mu} \left\lbrack {{Formula}\mspace{14mu} 242} \right\rbrack} & \; \\ {{\underset{m_{p} \times m_{p}}{\left\lbrack \lambda_{p} \right\rbrack}\underset{m_{p} \times 1}{\left\{ a_{p} \right\}}} = {{\underset{m_{p} \times n_{h}^{*}}{\left\lbrack e_{p}^{*} \right\rbrack^{T}}{\int_{S}{\underset{n_{h}^{*} \times 2}{\left\lbrack \psi_{h}^{*} \right\rbrack^{T}}\underset{2 \times 1}{\left\{ f \right\}}{s}}}} - {\underset{m_{p} \times n_{h}^{*}}{\left\lbrack e_{p}^{*} \right\rbrack^{T}}{\int_{S}{\underset{n_{h}^{*} \times 2}{\left\lbrack \psi_{p}^{*} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{b}}{\left\lbrack \chi_{B} \right\rbrack}{s}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}}}}}} & (427) \end{matrix}$

Rearrangement of the equation gives:

$\begin{matrix} {\mspace{20mu} \left\lbrack {{Formula}\mspace{14mu} 243} \right\rbrack} & \; \\ {\underset{m_{p} \times 1}{\left\{ a_{p} \right\}} = {\underset{m_{p} \times m_{p}}{\left\lbrack \lambda_{p} \right\rbrack^{- 1}}{\underset{m_{p} \times n_{h}^{*}}{\left\lbrack e_{p}^{*} \right\rbrack}}^{T}\left( {{\int_{S}{\underset{n_{h}^{*} \times 2}{\left\lbrack \psi_{h}^{*} \right\rbrack^{T}}\underset{2 \times 1}{\left\{ f \right\}}{s}}} - {\int_{S}{\underset{n_{h}^{*} \times 2}{\left\lbrack \psi_{h}^{*} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{b}}{\left\lbrack \chi_{B} \right\rbrack}{s}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}}}} \right)}} & (428) \end{matrix}$

Using eqs. (293), (310), and (373), we calculate the integration result of the right side of the equation as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 244} \right\rbrack & \; \\ \begin{matrix} {\underset{n_{h}^{*} \times 1}{\left\{ C_{F} \right\}} \equiv {\int_{S}{\underset{n_{h}^{*} \times 2}{\left\lbrack \psi_{h}^{*} \right\rbrack^{T}}\underset{2 \times 1}{\left\{ f \right\}}{s}}}} \\ {= {\underset{n_{h}^{*} \times {({n + \lambda})}}{\left\lbrack H_{h}^{*} \right\rbrack^{T}}{\int_{S}{\underset{({n + \lambda})}{\lbrack\Gamma\rbrack^{T}}\underset{2 \times 1}{\left\{ f \right\}}{s}}}}} \\ {= {\underset{n_{h}^{*} \times {({n + \lambda})}}{\left\lbrack H_{h}^{*} \right\rbrack^{T}}{\int_{S}{\begin{bmatrix} \underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack^{T}} \\ \underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}} \end{bmatrix}\underset{2 \times 1}{\left\{ f \right\}}{s}}}}} \end{matrix} & (429) \\ \begin{matrix} {\underset{n_{h}^{*} \times n_{b}}{\left\lbrack C_{B} \right\rbrack} \equiv {\int_{S}{\underset{n_{h}^{*} \times 2}{\left\lbrack \psi_{h}^{*} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{b}}{\left\lbrack \chi_{B} \right\rbrack}{s}}}} \\ {= {\underset{n_{h}^{*} \times {({n + \lambda})}}{\left\lbrack H_{h}^{*} \right\rbrack}{\int_{S}{\underset{{({n + \lambda})} \times 2}{\lbrack\Gamma\rbrack}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times {({n + \lambda})}}{\lbrack\Gamma\rbrack}{s}\underset{{({n + \lambda})} \times n_{b}}{\left\lbrack H_{b} \right\rbrack}}}}} \\ {= {\underset{n_{h}^{*} \times {({n + \lambda})}}{\left\lbrack H_{h}^{*} \right\rbrack^{T}}{\int_{S}{\begin{bmatrix} \underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack^{T}} \\ \underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}} \end{bmatrix}{\underset{2 \times 2}{\lbrack L\rbrack}\begin{bmatrix} \underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack} & \underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack} \end{bmatrix}}{s}\underset{{({n + \lambda})} \times n_{b}}{\left\lbrack H_{b} \right\rbrack}}}}} \\ {= {\underset{n_{h}^{*} \times {({n + \lambda})}}{\left\lbrack H_{h}^{*} \right\rbrack^{T}}{\int_{S}{\begin{bmatrix} {\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}} & {\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \\ {\underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}} & {\underset{\lambda \times 2}{\left\lbrack \psi_{o} \right\rbrack^{T}}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}} \end{bmatrix}{s}\underset{{({n + \lambda})} \times n_{b}}{\left\lbrack H_{b} \right\rbrack}}}}} \end{matrix} & (430) \end{matrix}$

Then, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 245} \right\rbrack & \; \\ {\underset{m_{p} \times 1}{\left\{ a_{p} \right\}} = {\underset{m_{p} \times m_{p}}{\left\lbrack \lambda_{p} \right\rbrack^{- 1}}\underset{m_{p} \times n_{h}^{*}}{\left\lbrack e_{p}^{*} \right\rbrack^{T}}\left( {\underset{n_{h}^{*}}{\left\{ C_{F} \right\}} - {\underset{n_{h}^{*} \times n_{b}}{\left\lbrack C_{B} \right\rbrack}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}}} \right)}} & (431) \end{matrix}$

We find that the coefficient vector {a_(p)} is settled corresponding to the known part s_(b).

When this equation is returned to eq. (424), in the case where no zero eigenvalue exists, the unknown parts s_(h) is settled. This results in that the problem is solved completely. In the case where a zero eigenvalue exists, the unknown parts s_(h) is a function of the coefficient vector {a₀}. Therefore, by adding the information on the boundary of the element, in addition to the node information, {a₀} can be determined.

In this way, when the unknown part s_(h) is determined, the unknown part s_(e) is determined according to eq. (300). Therefore, by eq. (299) or (301), the displacement {u} is determined completely. When the displacement is determined, the internal stress distribution, strain energy distribution, etc. of the element can be obtained.

8.18 Handling of Zero Eigenvalue

The eigenvalue is zero in the case where, in the primal and dual simultaneous differential equations of eqs. (384) and (385), either one of the following equations is established:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 246} \right\rbrack & \; \\ {{\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times 1}{\left\{ \varphi \right\}}} = \underset{2 \times 1}{\left\{ 0 \right\}}} & (432) \\ \left\lbrack {{Formula}\mspace{14mu} 247} \right\rbrack & \; \\ {{\underset{2 \times 2}{\left\lbrack L^{*} \right\rbrack}\underset{2 \times 1}{\left\{ \varphi^{*} \right\}}} = \underset{2 \times 1}{\left\{ 0 \right\}}} & (433) \end{matrix}$

In eqs. (415) and (423), eq. (432) is assumed to be established. As this means a homogenized differential equation, the coefficient {a₀} accompanying the solution thereof is arbitrary. The previous section shows the method of determining the arbitrarity thereof by adding the information on the boundary of the element, but there is another method in which a designer that performs the calculation determines it for him/herself. A system that displays in diagrams what deformation state or what stress state is taken according to the coefficient {a₀} is provided, and viewing the diagrams, the designer determines the coefficient {a₀} for him/herself so that a practical boundary condition is provided.

9. Use of Conventional Finite Element 9.1 Equation of Whole System

In Section (4) of Chapter 7, it is described that with respect to the conventional linear elastic finite element, particularly with respect to the h-element, an operation on a variation is not performed, and the obtained equation coincides with the energy conservation law of eq. (134). By combining a plurality of h-elements, we obtain an equation of motion of a whole system as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 248} \right\rbrack & \; \\ {{\underset{n \times n}{\lbrack K\rbrack}\underset{n \times 1}{\left\{ U \right\}}} = \underset{n \times 1}{\left\{ F \right\}}} & (434) \end{matrix}$

The solution method is constructed on the premise that only either a displacement or an external force of each node is given, that is, on the premise that a self-adjoint boundary condition is given. For example, it is assumed that among n degrees of freedom in total, n₁ degrees of freedom belong to the field such that the displacement is known, and n₂ degrees of freedom belong to the field such that the external force is known, then we obtain:

[Formula 249]

n=n ₁ +n ₂  (435)

By rearranging components of a stiffness matrix [K], a nodal displacement {U}, and a nodal external force {F}, we express eq. (434) as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 250} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n_{1} \times n_{1}}{\left\lbrack K_{11} \right\rbrack} & \underset{n_{1} \times n_{2}}{\left\lbrack K_{12} \right\rbrack} \\ \underset{n_{2} \times n_{1}}{\left\lbrack K_{21} \right\rbrack} & \underset{n_{2} \times n_{2}}{\left\lbrack K_{22} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n_{1} \times 1}{\left\{ U_{1} \right\}} \\ \underset{n_{2} \times 1}{\left\{ U_{2} \right\}} \end{Bmatrix}} = \begin{Bmatrix} \underset{n_{1} \times 1}{\left\{ F_{1} \right\}} \\ \underset{n_{2} \times 1}{\left\{ F_{2} \right\}} \end{Bmatrix}} & (436) \end{matrix}$

Known quantities are {U₁} and {F₂}, and unknown quantities are {U₂} and {F₁}. By separating eq. (436), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 251} \right\rbrack & \; \\ {{{{\underset{n_{1} \times n_{1}}{\left\lbrack K_{11} \right\rbrack}\underset{n_{1} \times 1}{\left\{ U_{1} \right\}}} + {\underset{n_{1} \times n_{2}}{\left\lbrack K_{12} \right\rbrack}\underset{n_{2} \times 1}{\left\{ U_{2} \right\}}}} = \underset{n_{1} \times 1}{\left\{ F_{1} \right\}}}{and}} & (437) \\ \left\lbrack {{Formula}\mspace{14mu} 252} \right\rbrack & \; \\ {{{\underset{n_{2} \times n_{1}}{\left\lbrack K_{21} \right\rbrack}\underset{n_{1} \times 1}{\left\{ U_{1} \right\}}} + {\underset{n_{2} \times n_{2}}{\left\lbrack K_{22} \right\rbrack}\underset{n_{2} \times 1}{\left\{ U_{2} \right\}}}} = \underset{n_{2} \times 1}{\left\{ F_{2} \right\}}} & (438) \end{matrix}$

Then, transforming eq. (438), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 253} \right\rbrack & \; \\ {\underset{n_{2} \times 1}{\left\{ U_{2} \right\}} = {\underset{n_{2} \times n_{2}}{\left\lbrack K_{22} \right\rbrack^{- 1}}\left( {\underset{n_{2} \times 1}{\left\{ F_{2} \right\}} - {\underset{n_{2} \times n_{1}}{\left\lbrack K_{21} \right\rbrack}\underset{n_{1} \times 1}{\left\{ U_{1} \right\}}}} \right)}} & (439) \end{matrix}$

Thus, the unknown amount {U₂} is settled. By substituting this into eq. (437), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 254} \right\rbrack & \; \\ {\underset{n_{1} \times 1}{\left\{ F_{1} \right\}} = {{\underset{n_{1} \times n_{1}}{\left\lbrack K_{11} \right\rbrack}\underset{n_{1} \times 1}{\left\{ U_{1} \right\}}} + {\underset{n_{1} \times n_{2}}{\left\lbrack K_{12} \right\rbrack}{\underset{n_{2} \times n_{2}}{\left\lbrack K_{22} \right\rbrack}}^{- 1}\left( {\underset{n_{2} \times 1}{\left\{ F_{2} \right\}} - {\underset{n_{2} \times n_{1}}{\left\lbrack K_{21} \right\rbrack}\underset{n_{1} \times 1}{\left\{ U_{1} \right\}}}} \right)}}} & (440) \end{matrix}$

Thus, the unknown amount {F₁} is settled. In other words, all of the unknown quantities are obtained. Such a procedure is established as a conventional practice of the finite element method.

However, eq. (434) is obtained from the energy conservation law, as an equivalent equation, without any variation operation being involved. This indicates that it is not necessary that the self-adjoint boundary condition should be imposed. Eq. (434) may be solved with respect to the non-self-adjoint boundary condition.

9.2 Calculation Under Non-Self-Adjoint Boundary Condition

With use of the unit matrix [I], the equation of motion (434) of the whole system is recognized as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 255} \right\rbrack & \; \\ {{\underset{n \times n}{\lbrack K\rbrack}\underset{n \times 1}{\left\{ U \right\}}} = {\underset{n \times n}{\lbrack I\rbrack}\underset{n \times 1}{\left\{ F \right\}}}} & (441) \end{matrix}$

Then, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 256} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n}{\lbrack K\rbrack} & \underset{n \times n}{\left\lbrack {- I} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n \times 1}{\left\{ U \right\}} \\ \underset{n \times 1}{\left\{ F \right\}} \end{Bmatrix}} = \underset{n \times 1}{\left\{ 0 \right\}}} & (442) \end{matrix}$

Combining the known part U_(b) of the nodal displacement U and the known part F_(b) of the nodal external force F, we define the nodal known part s_(b) as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 257} \right\rbrack & \; \\ {\underset{n_{b} \times 1}{\left\{ s_{b} \right\}} \equiv \begin{Bmatrix} \underset{n_{U_{b}} \times 1}{\left\{ U_{b} \right\}} \\ \underset{n_{F_{b}} \times 1}{\left\{ F_{b} \right\}} \end{Bmatrix}} & (443) \end{matrix}$

Combining the unknown part U_(v) of the nodal displacement U and the unknown part F_(v) of the nodal external force F, we define the nodal unknown part s_(v) as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 258} \right\rbrack & \; \\ {\underset{n_{v}}{\left\{ s_{v} \right\}} \equiv \begin{Bmatrix} \underset{n_{U_{v} \times 1}}{\left\{ U_{v} \right\}} \\ \underset{n_{F_{v\;} \times 1}}{\left\{ F_{v} \right\}} \end{Bmatrix}} & (444) \end{matrix}$

Here, the following is given:

[Formula 259]

n _(b) +n _(v)=2n  (445)

When U and F in eq. (442) are rearranged in the order according to eqs. (443) and (444), the columns of the matrix parts are also rearranged as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 260} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n_{b\;}}{\left\lbrack K_{b} \right\rbrack} & \underset{n \times n_{v}}{\left\lbrack K_{v} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n_{b} \times 1}{\left\{ s_{b} \right\}} \\ \underset{n_{v} \times 1}{\left\{ s_{v} \right\}} \end{Bmatrix}} = \underset{n \times 1}{\left\{ 0 \right\}}} & (446) \end{matrix}$

Here, the matrix [K_(b)] is obtained by extracting corresponding columns from the matrices [K] and [−I] in the order in which the known parts U_(b) and F_(b) are arranged, and similarly, the matrix [K_(v)] is obtained by extracting corresponding columns from the matrix [K] and [−I] in the order in which the unknown parts U_(v) and F_(v) are arranged. From eq. (446), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 261} \right\rbrack & \; \\ {{\underset{n \times n_{v\;}}{\left\lbrack K_{v} \right\rbrack}\underset{n_{v} \times 1}{\left\{ s_{v} \right\}}} = {{- \underset{n \times n_{b\;}}{\left\lbrack K_{b} \right\rbrack}}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}}} & (447) \end{matrix}$

Since the right side of the equation is composed of the known parts alone, it is possible to obtain the nodal unknown part s_(v) on the left side of the equation. In other words, depending on whether the matrix [K_(v)] is an underdetermined system in which a plurality of solutions exist or it is an overdetermined system in which no solution possibly exists, the unknown part s_(v) can be obtained based on the knowledge of linear algebra. In the case where a plurality of solutions exist, this case corresponds to the zero eigenvalue case described in Section 8.18, a designer him/herself can determine a solution.

9.3 Case with Gravity Load

The nodal external force F of the equation of motion (434) of the whole system is divided into a term F_(g) that represents a body force such as gravity load, and a term F_(s) that represents a surface force, and is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 262} \right\rbrack & \; \\ {\underset{n \times 1}{\left\{ F \right\}} \equiv {\underset{n \times 1}{\left\{ F_{g} \right\}} + \underset{n \times 1}{\left\{ F_{s} \right\}}}} & (448) \end{matrix}$

Then, eq. (434) is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 263} \right\rbrack & \; \\ {{\underset{n \times n}{\lbrack K\rbrack}\underset{n \times 1}{\left\{ U \right\}}} = {\underset{n \times 1}{\left\{ F_{g} \right\}} + \underset{n \times 1}{\left\{ F_{s} \right\}}}} & (449) \end{matrix}$

Transposing the surface force F_(s), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 264} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n}{\lbrack K\rbrack} & \underset{n \times n}{\left\lbrack {- I} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n \times 1}{\left\{ U \right\}} \\ \underset{n \times 1}{\left\{ F_{s} \right\}} \end{Bmatrix}} = \underset{n \times 1}{\left\{ F_{g} \right\}}} & (450) \end{matrix}$

Through the same procedure as that for obtaining eq. (446), this equation is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 265} \right\rbrack & \; \\ {{\begin{bmatrix} \underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack} & \underset{n \times n_{v}}{\left\lbrack K_{v} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n_{b} \times 1}{\left\{ s_{b} \right\}} \\ \underset{n_{v} \times 1}{\left\{ s_{v} \right\}} \end{Bmatrix}} = \underset{n \times 1}{\left\{ F_{g} \right\}}} & (451) \end{matrix}$

By transforming this equation so that the unknown part remains on the left side of the equation, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 266} \right\rbrack & \; \\ {{\underset{n \times n_{v}}{\left\lbrack K_{v} \right\rbrack}\underset{n_{v} \times 1}{\left\{ s_{v} \right\}}} = {{{- \underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack}}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}} + \underset{n \times 1}{\left\{ F_{g} \right\}}}} & (452) \end{matrix}$

As is the case with the case described in the previous section, based on the knowledge of linear algebra, we can obtain the unknown part s_(v).

9.4 Case where a Plurality of Solutions Exist

In the case where a plurality of solutions exist, one particular solution {s_(p)} that satisfies eq. (452), and homogeneous solutions {s₀} that is the solution when the right side of eq. (452) is given as zero, are obtained. The number no of the homogeneous solution is given as:

[Formula 267]

n _(o) ≡n _(v) −n  (453)

Let a matrix in which the homogeneous solutions {s₀} are arranged in the column direction (horizontal direction) be [ψ₀], which is as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 268} \right\rbrack & \; \\ {\underset{n_{v} \times n_{o}}{\left\lbrack \psi_{o} \right\rbrack} \equiv \begin{bmatrix} \underset{1}{\left\{ s_{o} \right\}} & \underset{2}{\left\{ s_{o} \right\}} & \Lambda & \underset{n_{o}}{\left\{ s_{o} \right\}} \end{bmatrix}} & (454) \end{matrix}$

Let a mode coefficient with respect to the homogeneous solution be {a₀}. The solution {s_(v)} is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 269} \right\rbrack & \; \\ {\underset{n_{v} \times 1}{\left\{ s_{v} \right\}} = {\underset{n_{v} \times 1}{\left\{ s_{p} \right\}} + {\underset{n_{v} \times n_{o}}{\left\lbrack \psi_{o} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o\;} \right\}}}}} & (455) \end{matrix}$

Now that all of the nodal displacement U and the nodal external force F are determined, the non-self-adjoint problem is solved. Both U an F are in the format including the mode coefficient {a₀}, then they are the solution with respect to an arbitrary mode coefficient {a₀}.

However, the mode coefficient {a₀} should be appropriately selected. Otherwise, with a conventional-type finite element, the solution sometimes indicates a state in which the element is folded and is kinked, and such analysis and result were often regarded as incorrect in many cases. This attributes to that, eq. (434), which is equivalent to the energy conservation law is satisfied as a whole system, but in an element, the differential equation is not satisfied.

Then, with focusing on that the deformation of the whole system is governed the mode coefficient {a₀}, the variation principle equivalent to the differential equation of the element, that is, the least-squares method of eqs. (122) to (128), is used, so that an appropriate {a₀} is determined. In the case where a state in which the element is folded and is kinked is not assumed at values around the appropriate {a₀}, it can be considered that this means that a sufficiently accurate analysis result is obtained.

9.5 Use of Least-Squares Method

In the case of a conventional-type finite element, an internal displacement of the element is represented by {u_(A)} of eq. (166). {u_(A)} is expressed with a nodal displacement U, and U is expressed with a mode coefficient {a₀}, as in eq. (455).

As the inside of the element is dominated by the differential equation, it is desirable that the mode coefficient {a₀} can be determined by substituting {u_(A)} into {u} of eq. (421). By creating a functional of eq. (128) with respect to the whole system and differentiating the same by the mode coefficient {a₀}, eq. (127) can be formed. As a result, the mode coefficient {a₀} can be determined.

A configuration may be such that the mode coefficient {a₀} is made in synchronization with a slider and a diagram for deformation or stress distribution is displayed in response to adjustment of the mode coefficient made by a designer using the slider. This allows the designer to study what supporting method is available, and allows the designer to have more images. This makes it possible to narrow down the state of feasible deformation and to re-define the boundary condition, thereby allowing the designer to make studies in order to obtain a better structure.

Hereinafter, embodiments of the present invention are described in detail, with reference to the drawings.

Embodiment 1

FIG. 2 is a functional block diagram showing an exemplary configuration of an analyzer of the present embodiment. The analyzer 10 (analysis system 10) shown in FIG. 2 can be formed with a computer that includes a calculation unit 1. The calculation unit 1 inputs design data of an analysis object as well as differential equation data including boundary condition data and differential operators with respect to the analysis object; calculates solutions of the differential equations of the analysis object; and outputs the same as analysis result data. Here, the calculation unit 1 calculates a solution u_(j) by the following equation where a variation of a dual displacement u_(i)* is given as a dual variation δu_(i)*. The following equation is the same as eq. (122).

$\begin{matrix} {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}^{*}{s}}}} = 0} & \left\lbrack {{Formula}\mspace{14mu} 1} \right\rbrack \end{matrix}$

For example, the calculation unit 1 inputs, as design data, data that show the shape and material of an analysis object, and reads data of an original differential operator L_(ij) to be used in the calculation, based on the design data. By using the original differential operator L_(ij) the calculation unit 1 calculates a solution of a differential equation expressed as the foregoing equation of Formula 1. This makes it possible to calculate a solution of a non-self-adjoint problem.

In the foregoing equation (Formula 1), as an example, f_(i) is assumed to represent a force acting on an analysis object, u_(i) is assumed to be a displacement of the analysis object, but f_(i) and u_(i) are not limited to these. Depending on an analysis object and a differential operator, f_(i) and u_(i) may represent something other than a force and a displacement in some cases. For example, in a variety of analysis such as structural analysis, vibration analysis, electric field analysis, material analysis, fluid analysis, thermal analysis, acoustic field analysis, electromagnetic analysis, circuit simulator, and the like, appropriate values for an analysis object can be set for f_(i) and u_(i).

Modification Example 1

The calculation unit 1 may calculate a solution u_(j) by the following equation. The following equation is identical to the above-described eq. (123). In this case, a solution of a self-adjoint problem can be calculated.

$\begin{matrix} {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}^{*}{s}}}} = 0} & \left\lbrack {{Formula}\mspace{14mu} 2} \right\rbrack \end{matrix}$

Modification Example 2

The calculation unit 1 can calculate a solution u_(j) of an analysis object by the following equation, by the direct variational method. The following equation is identical to the above-described eq. (127).

$\begin{matrix} {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{i}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}{\sum\limits_{j}{L_{ij}\; u_{i}^{*}{s}}}}}} = 0} & \left\lbrack {{Formula}\mspace{14mu} 3} \right\rbrack \end{matrix}$

In the calculation of a solution u_(j) by the direct variational method, a solution such that a variation is zero may be calculated in the functional Π of the following equation.

$\begin{matrix} {\Pi \equiv {\sum\limits_{i}{\int_{S}{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right)^{2}{{s}.}}}}} & \left\lbrack {{Formula}\mspace{14mu} 4} \right\rbrack \end{matrix}$

Embodiment 2

FIG. 3 is a functional block diagram showing an exemplary configuration of an analyzer of Embodiment 2. An analyzer 10 a shown in FIG. 3 includes a setting unit 11, an adjoint boundary condition calculation unit 12, and a non-self-adjoint calculation unit 13.

The setting unit 11 sets an original differential operator and a boundary condition of a displacement of an analysis object. For example, the setting unit 11 is capable of receiving data that show a configuration or a material of an analysis object, as design data from a user, and determining an original differential operator based on the configuration or the material of the analysis object. Further, the setting unit 11 can receive input of boundary condition data from the user.

The adjoint boundary condition calculation unit 12 calculates an adjoint boundary condition based on the boundary condition received by the setting unit 11. For example, the adjoint boundary condition calculation unit 12 is capable of calculating an adjoint boundary condition from a boundary term that is obtained by partial integration of a sum of integration, that is, an inner product, obtained by multiplying a differential equation that includes an original differential operator contained in differential equation data that the setting unit 11 has read, by a dual eigenfunction φ_(i)* or a dual displacement u_(i)*.

For example, in the case where the analysis object is a two-dimensional elastic body, it is possible to calculate an adjoint boundary condition by using the above-described eq. (91). In this case, in eq. (91), such a condition for φ_(i)* and p_(Ei)* that a boundary term is zero in the case where a homogeneous boundary condition imposed on u_(Hi) and p_(Hi) is imposed on φ_(i) and p_(Ei) can be determined to be an adjoint boundary condition. As one example, in the case where a boundary condition such that a force is zero (p_(Ei)=0) is imposed on a boundary surface of an analysis object, φ_(i)* may be arbitrary and either p_(Ei)* or φ_(i) has to be zero so that a boundary term R_(E) is zero. Here, in the case where a boundary condition that a displacement is also zero (φ_(i)=0) is imposed on the boundary surface of the analysis object, p_(Ei)* is arbitrary. In this case, in the adjoint boundary condition, both of the displacement and the force of the boundary surface are arbitrary. In such a case, the boundary condition and the adjoint boundary condition do not coincide, the problem is a non-self-adjoint problem. However, according to the effects of eq. (106), Σ_(j)L_(ji)*φ_(j)* has to satisfy the boundary condition imposed on φ_(i). This is included in the dual boundary condition. In the case of a non-self-adjoint problem, the problem is solved appropriately, by using the primal boundary condition and the dual boundary condition.

The non-self-adjoint calculation unit 13 calculates primal simultaneous differential equations and dual simultaneous differential equations from original differential operators, and determine a primal eigenfunction and a dual eigenfunction by using the primal simultaneous differential equations and the dual simultaneous differential equations, as well as the boundary condition and the adjoint boundary condition, thereby calculating solutions of simultaneous differential equations.

For example, let an original differential operator be L_(ij). Then, a primal simultaneous differential equation can be defined as eq. (105), and a dual simultaneous differential equation can be defined as eq. (106). Further, according to eq. (90), an adjoint differential operator L_(ij)* can be calculated. The non-self-adjoint calculation unit 13 can defines a primal simultaneous eigenvalue problem (eq. (98)) and a dual simultaneous eigenvalue problem (eq. (99)) by using the primal simultaneous differential equation (eq. (105)), the dual simultaneous differential equation (eq. (106)), the boundary condition, and the adjoint boundary condition, and can calculate a primal eigenfunction φ, and a dual eigenfunction φ*.

It should be noted that the solution method used by the non-self-adjoint calculation unit 13 is not limited to the solution method using the primal simultaneous eigenvalue problem (eq. (98)) and the dual simultaneous eigenvalue problem (eq. (99)). For example, it is possible to calculate a primal eigenfunction φ, and a dual eigenfunction φ* by using eqs. (122) and (123). In this case, the primal eigenfunction φ and the dual eigenfunction φ* can be calculated analytically and approximately. For example, as is described below, an approximate eigenfunction can be calculated by using eqs. (122) and (123).

The non-self-adjoint calculation unit 13 calculates a solution of simultaneous equations of a primal problem by using a primal eigenfunction φ and a dual eigenfunction φ*. For example, it is capable of calculating a coefficient c_(k) expressed by eq. (111), and substituting the same into eq. (69), to calculate a solution function u_(Hj). Further, non-self-adjoint calculation unit 13 is capable of calculating a coefficient c_(k)* of a dual problem according to eq. (115), and substituting the same into eq. (70), to calculate a solution function u_(Hj)* of the dual problem.

Operation example

FIG. 4 is a flowchart showing an exemplary operation of the analyzer 10 a shown in FIG. 3. In the example shown in FIG. 4, the setting unit 11 receives input of design data from a user (S1). The design data are not limited to a specific object or format. For example, in the case where the analysis object is a solid matter, the setting unit 11 can receive input of data representing a shape of a material of the analysis object. In the case where the analysis object is a fluid, the setting unit 11 can receive input of data representing physical properties of the fluid. Further, in the case of analysis by the finite element method, the setting unit 11 may receive input of data showing a configuration or physical properties of each element. Further, the setting unit 11 is capable of, not only receiving input from a user, but also obtaining design data by, for example, reading the design data from an accessible recording medium, or downloading the same via a network.

The setting unit 11 determines an original differential operator based on design data obtained at S1, and determine a differential equation of a primal problem (S2). For example, the setting unit 11 is capable of extracting differential operators corresponding to a shape of the analysis object indicated by design data from preliminarily recorded data, and setting differential equations including the extracted differential operators as differential equations of the primal problem. In this way, it is possible to determine differential equations of a primal problem in accordance with design data. Here, as one example, a case where the original differential operator is explained.

The setting unit 11 further obtains boundary condition data (S3). The boundary condition data are data that relate to a boundary condition of an analysis object, and are not limited to a specific object or format. For example, data indicating a condition of at least one of a displacement and a force of an analysis object on a boundary surface can be input as a boundary condition. In the present invention, both of a displacement and a force can be set, which could not been set conventionally.

The adjoint boundary condition calculation unit 12 generates adjoint boundary condition data, using the differential equations set at S2, and the boundary condition data obtained at S3. The adjoint boundary condition may be calculated by using eq. (91), as described above. Further, in the exemplary case where the finite element method is used, the boundary term expressed by eq. (60) can be calculated by using eqs. (208), (229), (207), and (228), which indicate a boundary displacement, a dual displacement, a surface force, and a dual surface force, respectively.

The non-self-adjoint calculation unit 13 generates data that indicate a primal simultaneous differential equation (for example, eq. (105)) and a dual simultaneous differential equation (eq. (106)), from the original differential operator L_(ij) (S5). Then, the non-self-adjoint calculation unit 13 calculates a primal eigenfunction φ, and a dual eigenfunction φ*, by using the primal simultaneous differential equation and the dual simultaneous differential equation calculated at S5, as well as the boundary condition obtained at S2 and the adjoint boundary condition calculated at S4 (S6).

By using the primal eigenfunction φ and the dual eigenfunction φ* calculated at S6, the non-self-adjoint calculation unit 13 calculates a solution function u_(Hj) of simultaneous equations of the primal problem, by the eigenfunction method, for example, by eqs. (111) and (69) (S7). Further, by using the primal eigenfunction φ, and the dual eigenfunction φ*, the non-self-adjoint calculation unit 13 can calculate a solution function u_(Hj)* of simultaneous equations of the dual problem, for example, by eqs. (115) and (70). Regarding the solution calculated in this way, an analysis result is output to a display of a computer (not shown) or the like (S8). A specific example of an analysis result is described below.

The above-described processing makes it possible to determine an adjoint boundary condition based on a boundary condition, and further, by calculating a primal differential operator and a dual differential operator from an original differential operator, determine a primal eigenfunction and a dual eigenfunction of these. By calculating a solution of simultaneous equations of the primal problem by using these eigenfunctions, a solution of the non-self-adjoint problem can be calculated. It should be noted that even regarding a self-adjoint problem, a solution thereof can be calculated through the foregoing processing.

So far, an exemplary operation of the analyzer 10 a has been described, but the processing by the analyzer 10 a is not limited to the above-described example. It should be noted that the present embodiment can be regarded as a specific example of Embodiment 1 described above.

Embodiment 3

FIG. 5 is a functional block diagram illustrating an exemplary configuration of an analyzer according to Embodiment 3. An analyzer 10 b shown in FIG. 5 includes a setting unit 11, an adjoint boundary condition calculation unit 12, a self-adjoint determination unit 15, a non-self-adjoint calculation unit 13, and a self-adjoint calculation unit 14. The setting unit 11, the adjoint boundary condition calculation unit 12, and the non-self-adjoint calculation unit 13 can be formed to have the same configurations as those in Embodiment 1 described above.

The analyzer 10 b includes a self-adjoint determination unit 15 that determines whether a boundary condition and an adjoint boundary condition calculated by the adjoint boundary condition calculation unit coincide. In the case where the self-adjoint determination unit 15 determines that the boundary condition and the adjoint boundary condition coincide, the self-adjoint calculation unit 14 determines a self-adjoint eigenfunction of the self-adjoint problem from original differential operators, so as to calculate a solution of the self-adjoint problem. In the case where it is determined that the boundary condition and the adjoint boundary condition do not coincide, the non-self-adjoint calculation unit 13 obtains a primal eigenfunction and a dual eigenfunction by using primal simultaneous differential equations and dual simultaneous differential equations, as well as the boundary condition and the adjoint boundary condition, so as to calculate solutions of simultaneous differential equations.

The self-adjoint determination unit 15 is capable of determining whether the boundary condition and the adjoint boundary condition are self-adjoint or non-self-adjoint, by determining whether the boundary condition and the adjoint boundary condition coincide. In other words, in the case where the original differential operator is self-adjoint and the boundary condition and the adjoint boundary condition coincide, they can be determined to be self-adjoint, and in the case where they do not coincide, they can be determined to be non-self-adjoint. Further, the determination by the self-adjoint determination unit 15 regarding whether the boundary condition and the adjoint boundary condition coincide or not involves the determination regarding whether the original differential operator and the adjoint differential operator L_(ji)* coincide or not (L_(ij)=L_(ji)* or not). For example, the adjoint differential operator L_(ij)* can be calculated by partial integration of a sum of integration, that is, an inner product, obtained by multiplying a differential equation including the original differential operator by a dual eigenfunction φ_(i)* or a dual displacement u_(i)*. For example, the adjoint differential operator L_(ij)* can be calculated by eq. (62) or (90).

The self-adjoint calculation unit 14 generates the simultaneous differential equations of the self-adjoint problem from the original differential operators, and calculates solutions of the simultaneous differential equations. For example, the self-adjoint calculation unit 14 can calculate the self-adjoint eigenfunction φ_(i) by using eq. (107) or (123) as simultaneous differential equations of the self-adjoint problem. The self-adjoint calculation unit 14, for example, calculates c_(k) by an equation in which the self-adjoint eigenfunction φ_(ik) is replaced with φ_(ik)* of eq. (111), and substituting this c_(k) into eq. (69), thereby obtaining a solution function u_(Hj) of the self-adjoint problem.

Operation Example

FIG. 6 is a flowchart showing an exemplary operation of the analyzer 10 b shown in FIG. 5. In FIG. 6, the analyzer 10 b can execute the processing of S1 to S4 in the same manner as the processing of S1 to S4 in Embodiment 2. At S21, the self-adjoint determination unit 15 compares data showing the boundary condition obtained at S1 and the adjoint boundary condition data generated at S4, and thereby determines whether the boundary condition coincides with the adjoint boundary condition or not. When the self-adjoint determination unit 15 determines at S21 that they do not coincide (i.e., non-self-adjoint), the non-self-adjoint calculation unit 13 executes the processing at S5 to S7. The processing at S5 to S7 may be executed in the same manner as that at S4 to S7 in Embodiment 2.

When the self-adjoint determination unit 15 determines that the boundary condition and the adjoint boundary condition coincide (i.e., self-adjoint) at S21, the self-adjoint calculation unit 14 generates self-adjoint differential equation data from the original differential operator (S22). For example, the self-adjoint calculation unit 14 can use eq. (107) as the self-adjoint differential equation. In this case, the self-adjoint calculation unit 14 calculates the self-adjoint eigenfunction φ_(i) by using the original differential operator L_(ij) obtained at S3 as a self-adjoint differential operator, and the boundary condition set at S2 as a self-adjoint boundary condition. Then, the self-adjoint calculation unit 14 calculates a solution function uj of the simultaneous differential equations of primal problem, by using the self-adjoint eigenfunction φ_(i) (S24). The result output at S8 may be executed in the same manner as that at S8 in Embodiment 2.

Through the above-described processing, whether a problem of an analysis object is self-adjoint or non-self-adjoint can be determined, and the processing can be switched appropriately according to the determination result. It should be noted that the present embodiment can be considered as a specific example of Embodiment 1 or 2 described above.

Embodiment 4

The present embodiment is an exemplary case where an analyzer is applied to an analysis using the finite element method (FEM). In the present embodiment, an original differential operator and a boundary condition are set with respect to a finite element. For example, the configuration and the analysis processing of the analyzer according to Embodiments 1 to 3 described above can be applied to a finite element as well.

Here, difference of the present embodiment, which is an embodiment in which the analysis processing according to the Embodiments 1 to 3 is used with respect to a finite element, from a conventional finite element method is explained. First of all, a variation operation is not executed in the conventional h-method as described above, while in the present embodiment, a solution is calculated by the direct variational method (for example, in Embodiment 1). The p-method is a technique for approximating an internal displacement of the element by a high-order function, in which a function that appropriately approximates an internal displacement of the element is found through trial and error and is used. In contrast, in the present embodiment, a primal eigenfunction and a dual eigenfunction are calculated, and with use of these, an internal displacement of the element is appropriately explained.

FIG. 7 is a flowchart showing an exemplary operation of an analyzer of the present embodiment. The analyzer of the present embodiment may have the same configuration as that shown in FIG. 2 or FIG. 5. It should be noted that the flowchart shown in FIG. 7 is merely one example, and the present invention is not limited to this. In the example shown in FIG. 7, first of all, the setting unit 11 receives input of data indicating a shape of an analysis object (S31). For example, data indicating a shape of a finite element are input. Further, the setting unit 11 creates a weight constant (S32). It is preferable that the weight constant w is used in such a manner that an eigenvalue λ is determined as a dimensionless quantity. For example, in the case where an analysis object is a ring, the order of the differential operator is second. Therefore, if the characteristic length is given as r, it is preferable that w=1/r² is set. Here, as the characteristic length, a length of a certain characteristic part of the object is used. In the case of a ring, a radius thereof is preferably used as a characteristic length. In this case, as eigenvalues λ of similar figures coincide, calculation can be omitted with respect to the object having a similar figure.

Further, the setting unit 11 inputs a nodal displacement and a nodal force (S33). For example, a displacement u and a dual displacement u* of a node can be expressed by a boundary function u_(A) that represent a displacement of an element boundary and a correction function u₀, and the correction function u₀ can be expressed by a sum of trial functions ψ₀ (For example, eqs. (173) and (190)). The setting unit 11 sets a trial function (S34). The trial function can be set, for example, as eq. (179). For example, in the case of 16-nodes quadrilateral element as shown in FIG. 8, a displacement on a boundary surface can be expressed by eq. (208), a dual displacement can be expressed by eq. (229), a surface force can be expressed by eq. (207), and a dual surface force can be expressed by eq. (228).

The adjoint boundary condition calculation unit 12 calculates an adjoint boundary condition from the boundary condition. For example, the adjoint boundary condition calculation unit 12 can calculate an adjoint boundary condition by substituting a displacement and a surface force of an element into a boundary term R that is obtained by partial integration of a sum of integration, that is, an inner product, obtained by multiplying differential equations that includes original differential operators by a dual displacement u_(i)*.

For example, the boundary term R can be expressed by eq. (250). In this case, in eq. (250), a matrix element of a nodal force F is divided into a known part F_(b) and an unknown part F_(v), and the same are rearranged, while a nodal displacement U is divided into a known part U_(b) and an unknown part U_(v), and the same are rearranged. A dual nodal force F* also is divided into a known part F_(b)* and an unknown part F_(v)*, and a dual displacement U* is divided into a known part U_(b)* and an unknown part U_(v)*, and the same are arranged so as to correspond to vector elements of the rearranged nodal forces F_(b), F_(y) and nodal displacements U_(b), U_(v). Then, a result like eq. (251) is obtained. In order to make R zero, U_(b)* and F_(b)* corresponding to the unknown parts F_(v) and U_(v), respectively, should be zero. Therefore, an adjoint boundary condition expressed by eqs. (254) and (255) are obtained.

The non-self-adjoint calculation unit 13, for example, generates a primal trial function [ψ_(h)] and a dual trial function [ψ_(h)*] (S37, S38). Let an approximate primal eigenfunction be {φ}, and let an approximate dual eigenfunction be {φ*}. Let their respective coefficient vectors be {e_(h)} and {e_(h)*}, and {φ} and {φ*} are given as:

{φ}=[ψ_(h)]{e_(h)} {φ*}=[ψ_(h)*]{e_(h)*}

Substituting these into the primal simultaneous differential equations (eq. (105)) and the dual simultaneous differential equations (eq. (106)), we obtain coefficient vectors {e_(h)} and {e_(h)*} according to eqs. (122) and (123), and thereafter, obtain approximate eigenfunctions {φ} and {φ*} (S39).

In one example, the primal trial function [ψ_(h)] can be expressed as eq. (313). In this case, a function matrix Γ is defined by eq. (293), and coefficient matrices [c_(e)] and [c_(h)] are defined by eqs. (294) and (295). The dual trial function [ψ_(h)*] can be expressed as, for example, eq. (373). In this case, coefficient matrices [c_(e)*] and [c_(h)*] are defined by eqs. (354) and (355).

The non-self-adjoint calculation unit 13 calculates a solution function of simultaneous equations of the primal problem by the eigenfunction method, by using the primal eigenfunction and the dual eigenfunction calculated at S39 (S40). Regarding the solution thus calculated, an analysis result is output to a computer display (not shown) or the like (S41). Specific examples of the analysis result are to be described below.

At S40, for example, according to (Formula 1), an equation (for example, eqs. (400), (401), or eq. (402)) obtained by multiplying the variation {δφ} of the approximate eigenfunction {φ} calculated at S37, and the variation {6V} of the approximate dual eigenfunction {V} calculated at S38 on the primal simultaneous differential equations and the dual simultaneous differential equations, and integrating the same, is solved, whereby an eigenvalue and an eigenvector can be obtained. With use of these, the displacement {u} of the primal problem is determined.

As described above, by using the finite element method, even in the case where the shape is complicated and it is difficult to calculate an eigenfunction directly, an approximate eigenfunction can be determined.

Embodiment 5

An information processing device 10 c illustrated in FIG. 67 includes an initial equation decision unit 21, a boundary condition decision unit 22, and a calculation unit 23. The initial equation decision unit 21 reads data that show a structure and properties of constituent elements of a system to be processed, and decides an equation that expresses the system. This equation may be, for example, n equations that include a variable representing a physical quantity to be determined. The equation can be expressed by, for example, data of a matrix or data of a differential equation, and the representation format of data is not limited particularly. The variable can be expressed by, for example, data of a vector, a function, or the like, but the representation format thereof is not limited particularly.

The initial equation decision unit 21 receives, for example, the designation of a system to be processed from a user, and can select simultaneous differential equations corresponding to the designated system, from preliminarily recorded simultaneous differential equations. Alternatively, the initial equation decision unit 21 may be configured to receive simultaneous differential equations from a user.

The boundary condition decision unit 22 reads a value representing a physical quantity as data representing a boundary condition, and decides a boundary condition. The boundary condition decision unit 22 also receives, for example, a boundary condition such that an external force and a displacement are designated with respect to one direction of one point, that is, 1 degree of freedom. Owing to the functions of the calculation unit 23, which are described below, even if a plurality of physical quantity are defined as a boundary condition with respect to 1 degree of freedom, such a case can be dealt with. As a boundary condition, not only designation of a known part but also designation of an unknown part can be received. For example, such a setting that the displacement and the external force are unknown with respect to one node can be received.

The calculation unit 23 transforms the n initial equations into an equation of 2n variables or an equation including 2n equations, then, decides a known part that includes a variable that becomes known according to a boundary condition, and an unknown part that includes an unknown variable, in the equation having 2n variables or the equation including 2n equations obtained by the transformation, and calculates a solution of the equation regarding the unknown part. For example, in the case where simultaneous differential equations are decided as initial equations, as shown in eq. (963), it can be transformed into a format having equations the number of which is twice the number of the initial equations. In the case where equations expressed by matrices and variable vectors are decided as initial equations, the initial equations can be transformed into, for example, as shown in eq. (442), matrices including variable vectors having a degree of freedom that is twice the degree of freedom of the variable vector of the initial equations.

The calculation unit 23 is capable of outputting information obtained from the calculated solution or information obtained from the solution. For example, in the case where an object is subjected to a force in a system to be processed and is displaced, the calculation unit 23 can output a transformed state, stress distribution, error distribution of the same, or the like, as information obtained from the solution.

(Exemplary Operation Using Adjoint Differential Operator)

The initial equation decision unit 21 is capable of deciding, for example, simultaneous differential equations that express the relationship between a force acting on an object as a constituent element of the system, and a variable that represents a physical quantity of the object. The physical quantity of the object may be, for example, a displacement, a velocity, or the like, but it is not limited specifically.

The calculation unit 23 generates data indicating the 2n equations, by using differential operators of simultaneous differential equations decided by the initial equation decision unit 21, and adjoint differential operators that are determined from the differential operators, and calculates solutions of the 2n equations. The calculation unit 23 newly introduces dual simultaneous equations (including n equations) that includes dual variables, with respect to differential operators and variables of simultaneous differential equations of a primal problem that represents a system to be processed. This results in calculation of solutions of 2n equations. In this way, adjoint differential operators that are determined from differential operators are used for generating the 2n equations, and solutions of the 2n equations are calculated. By doing so, even in the case where a plurality of physical quantities are set as a boundary condition with respect to 1 degree of freedom, solutions can be calculated.

It should be noted that, for example, as shown in eq. (64), an inner product of the result of the differential operators of the simultaneous differential equations of primal problem acting on the primal variables with the dual variables is equal to an inner product of the primal variables with a result of the differential operators of the dual simultaneous differential equations acting on the dual variables. In the present specification, various examples of determining dual differential equations regarding various simultaneous differential equations are disclosed. It should be noted that information about dual differential equations could be recorded in association with differential equations preliminarily. The calculation unit 23 can read out information about dual differential equations corresponding to differential equations decided by the initial equation decision unit 21, and can use the same in an operation.

(Exemplary Operation with Respect to Finite Element)

Further, the calculation unit 23 is capable of calculating a solution of an equation, without calculating an adjoint operator, by transforming initial equations into an equation that includes 2n variables. For example, the initial equation decision unit 21 decides two n-dimensional variable vectors and an n-row matrix, thereby deciding initial equations that indicate physical quantities at nodes of a constituent element of a system. For example, the calculation unit 23 is capable of generating an n-row matrix that represents the relationship between an n-dimensional vector of variables that represent displacements of nodes and an n-dimensional vector of variables that represent nodal forces. This can be considered to express differential equations that the system should satisfy, by using the n-dimensional vectors and the n-row matrix. For example, initial equations can be decided in the form as shown by eq. (441), eq. (449), or the like. In the example of eq. (441), n equations are expressed by an n×n matrix K and variable vectors {U} and {F} that are n-dimensional vectors.

The boundary condition decision unit 22 is configured to be able to decide a boundary condition in which, regarding variables of equations, the number of degrees of freedom of variables whose values are known, and the number of degrees of freedom of variables whose values are unknown, are different. For example, The boundary condition decision unit 22 may be configured so that even in the case where a user specifies a boundary condition such that a plurality of physical quantities are unknown or known with respect to one degree of freedom, an error display indicating that the setting by the user is inappropriate should not be output.

In this case, the calculation unit 23 generates a 2n-dimensional vector, based on two variable vectors. Further, based on variables of this 2n-dimensional vector, the matrix is transformed into a 2n-column matrix, and among the variables of the 2n-dimensional vector, a known part that includes variables that are made known by a boundary condition, and an unknown part that includes unknown variables, are decided. Here, the degree of freedom of the unknown part, and the degree of freedom of the known part do not have to be the same. For example, the calculation unit 23 can decide the unknown part and the known part, without imposing a condition that the degree of freedom of the unknown part and the degree of freedom of the known part should be the same. The calculation unit 23 transforms the 2n-column matrix and the 2n-dimensional vector so that a variable of the unknown part is expressed by a variable of the known part, and the variable of the unknown part is calculated with use of the transformed matrix. For example, the calculation unit 23 can rearrange elements of the variable vector so that the known part and the unknown part of each variable are combined, respectively, and accordingly, arrange the elements of the matrix. It should be noted that this specific example is shown by eqs. (441) to (447) in Section 9.2. The calculation unit 23 can calculate a solution by executing the foregoing processing, with respect to both of a self-adjoint problem and a non-self-adjoint problem. For example, the calculation unit 23 can accept the case where a boundary condition of 1 degree of freedom, for example, such that a displacement and a force in the x direction of a certain node are both zero is set, or the case where a boundary condition such that a displacement and a force in the y direction of a certain node are both unknown is set. Such an operation as described above, by an equation having 2 times of variables, without any limitation on the numbers of degrees of freedom of the unknown part and the known part, enables calculation of a solution even under a non-self-adjoint boundary condition.

In the case where a plurality of solutions exist, for example, it is possible to decide a mode coefficient with respect to homogeneous solutions by a functional IT of eq. (128) so that variations are zero, and determine a solution by using the decided mode coefficient. Further, the configuration may be such that input of a value in the vicinity of the decided mode coefficient is received from a user, a solution is calculated by using a mode coefficient of an input value, and the solution or information obtained from the solution is output. It should be noted that it is not necessary to provide such a configuration in an information processing device provided with the calculation unit 23. Alternatively, with respect to a plurality of solutions calculated by the calculation unit 23, the processing for determining a solution by using the mode coefficient may be executed by another information processing device. Further alternatively, the configuration may be such that input of a value in the vicinity of the mode coefficient from a user is received, and the processing of calculating a solution by using a mode coefficient of the input value and outputting the same is executed by another information processing device.

For example, in the case where as solutions of an equation calculated by the calculation unit 23, one particular solution and a plurality of homogeneous solution are obtained, the solutions can be expressed as the particular solution and the homogeneous solutions multiplied by an arbitrary mode coefficient. A set of mode coefficients can be determined by applying the solution expressed as the function of the mode coefficients to the equation (127) such that the variation of the functional IT of equation (128) should be zero. By using the set of mode coefficients determined herein, a solution that satisfies the differential equation approximately as a whole structural system can be obtained. Exemplary processing in such a case where a plurality of solutions exist are described in, for example, Sections 9.4, 9.5, and 11.3.10. In the present embodiment, the calculation unit 23 calculates a solution of an equation where the degree of freedom is increased to twice, in a situation in which the degree of freedom of an unknown part and the degree of freedom of a known part are not necessarily identical. With this calculation unit 23, a plurality of solutions are calculated in some cases. In such a case as well, as described above, a typical solution among the plurality of solutions is calculated, and outputting of a solution is executed in a range that includes this solution, whereby an appropriate result can be output. Further, an operation by a user causes a solution to vary and be displayed. Thus, a user interface that is capable of conveying the result of a plurality of solutions in a recognizable style is provided.

Embodiment 6

An information processing device 10 d shown in FIG. 68 includes an initial equation decision unit 21, a boundary condition decision unit 22, and a determination unit 33. The initial equation decision unit 21 and the boundary condition decision unit 22 can be configured in the same manner as that of the initial equation decision unit 21 and the boundary condition decision unit 22 shown in FIG. 67. The determination unit 33 determines whether the boundary condition of the simultaneous differential equations decided by the initial equation decision unit 21 is a self-adjoint condition or not.

(Example of Determination Using Adjoint Boundary Condition)

For example, in the case where the same number of dual variables as the number of variables of simultaneous differential equations and dual simultaneous differential equations are defined, the determination unit 33 calculates a boundary term obtained by partial integration of a sum of integration, that is, an inner product, of the simultaneous differential equations and the dual variables, and can decide such a condition of the dual variables that under a boundary condition decided by the boundary condition decision unit 22, the boundary term is zero, as an adjoint boundary condition. In the case where the adjoint boundary condition coincides with the boundary condition, the determination unit 33 determines that it is a self-adjoint condition.

In order to calculate the adjoint boundary condition, for example, equations for calculating a boundary term in association with differential equations can be recorded preliminarily in a recording section that is accessible by the information processing device 10 d. The determination unit 33 can read equations of a boundary term corresponding to the differential equations decided by the initial equation decision unit 21, from the recording section, and calculate a boundary term.

The following is a specific example. In a quadrilateral element model, as shown by eq. (250) or eq. (1814), a boundary term R can be expressed by a product of a transposed matrix of a nodal external force F and displacement U, and a matrix of a dual nodal external force F* and a dual displacement U*. In such a case, as shown by eq. (251) or eq. (1814), each of the nodal external force F, the displacement U, the dual nodal external force F* and the dual displacement U* is divided into an unknown part and a known part, and they are rearranged, whereby an adjoint boundary condition is obtained. In this way, in a boundary term expressed by a transposed matrix of variables of differential equations and a matrix of dual variables, each variable is divided into an unknown part and a known part, and they are rearranged, whereby an adjoint boundary condition can be obtained. For example, eq. (251) or eq. (1814) can be used in the case where operators and a boundary condition of initial differential equations are set with respect to a finite element so that analysis is executed.

The determination unit is, for example, capable of outputting information based on a determination result regarding whether or not it is a self-adjoint condition, to a user. For example, in the case where the boundary condition and the adjoint boundary condition do not coincide and it is determined to be a non-self-adjoint condition, the determination unit can output a message saying that the analysis object is non-self-adjoint under the boundary condition. For example, the determination unit 33 is capable of outputting information that guiding the analysis of a non-self-adjoint problem, such as information that a function of dealing with a non-self-adjoint problem is necessary for analysis, where software that enables analysis of a non-self-adjoint problem is available and how to obtain it, etc.

Alternatively, based on information about whether or not the adjoint boundary condition and the boundary condition coincide, the determination unit 33 is capable of controlling a subsequent operation. For example, in the case where the adjoint boundary condition and the boundary condition coincide, the determination unit 33 causes analysis processing using the conventional finite element method to be executed, and in the case where the adjoint boundary condition and the boundary condition do not coincide, as described above, the determination unit 33 causes the information processing device to execute processing that can deal with a non-self-adjoint problem. Processing that can deal with a non-self-adjoint problem is, for example, processing of transforming initial equations into an equation that includes 2n equations, and in the transformed equation including 2n equations, determining a known part including a known variable and an unknown part including an unknown variable, thereby calculating a solution of an equation with respect to the unknown part.

(Example of Determining Self-Adjoint Condition without Calculation of Adjoint Boundary Condition)

It should be noted that there are some cases where the determination of a self-adjoint condition can be achieved without the calculation of an adjoint boundary condition as in the foregoing case. For example, as described in Section 11.7.1, in the case where a self-adjoint boundary condition is achieved, according to eq. (251) or eq. (1816), the number “n1” of known parts Ub of a nodal displacement U is equal to the number of unknown parts Fv of a nodal external force, and the number “n2” of known parts Fb of a nodal external force F is equal to the number of unknown parts Uv of a nodal displacement. Therefore, the determination unit 33 can determine whether it is a self-adjoint condition or not, based on whether or not a degree of freedom of an unknown part of one variable is equal to a degree of freedom of a known part of another variable.

(Exemplary Combination of Determination Unit and Calculation Unit)

It should be noted that a configuration obtained by adding the determination unit 33 in FIG. 68 to the information processing device in FIG. 67 is possible. For example, based on the determination result by the determination unit 33, a control operation can be executed regarding whether or not the calculation unit 23 should perform an operation using the 2n-dimensional equation. With this, the calculation unit 23 can appropriately switch a non-self-adjoint operation using the 2n-dimensional equation, and a self-adjoint operation using the n-dimensional equation, thereby improving the processing efficiency.

The following is a specific example of the same. The boundary condition decision unit 22 can receive the designation of a known part and an unknown part of the variable from a user. Based on the designation received from the user, the determination unit 33 decides the unknown part and the known part of the variable, and outputs information about whether or not a degree of freedom of an unknown part of one variable is equal to a degree of freedom of a known part of another variable. In the case where the determination unit 33 determines that the degree of freedom of the unknown part of one variable is equal to the degree of freedom of the known part of another variable, the calculation unit 23 calculates the unknown part of the variable by using the n-row matrix expressing the differential operators. In the case where the determination unit 33 determines that the degree of freedom of the unknown part of one variable is not equal to the degree of freedom of the known part of another variable, the differential equations are transformed to a format having 2n variables and including 2n equations, and among the 2n transformed equations, solutions of the equations with respect to the unknown part is calculated.

Operation Example Exemplary Setting Applicable to Case of Self-Adjoint Condition

FIG. 69 is a flowchart showing exemplary processing for setting a nodal displacement vector, and a nodal external force vector, in the conventional analysis with respect to a finite element, on the premise that the boundary condition satisfies the self-adjoint condition. In the example shown in FIG. 69, the information processing device first receives input of a nodal displacement (S101). For example, a degree of freedom for which a displacement is known and its value are input by a user. Alternatively, data indicating a degree of freedom of a nodal displacement and its value may be read in. The degree of freedom is specified by, for example, the position and direction of a node. When a nodal displacement is input, a degree of freedom of a known part Ub of a nodal displacement and its value are settled (S102). When the degree of freedom of the known part Ub of the nodal displacement and its value are settled, a degree of freedom of an unknown part Fv of a nodal external force can be fixed automatically (S103). This is because in the case where a self-adjoint condition is satisfied, the degree of freedom of the unknown part Fv is equal to that of the known part Ub.

When the degree of freedom of the known part Ub of the nodal displacement is settled, the degree of freedom of the known part Ub is subtracted from a total degree of freedom, whereby the degree of freedom of the unknown part Uv of the nodal displacement is determined (S104). In the case where a self-adjoint condition is satisfied, the degree of freedom of the known part Fb is equal to the degree of freedom of the unknown part Uv. Therefore, the degree of freedom of the known part Fb is settled, whereby the value of the known part Fb is assumed to be zero (S105). At S107, when a degree of freedom and a value of a nodal external force are input by a user, the information processing device checks if the known part Fb of the nodal external force is not incompatible with the degree of freedom, and corrects the value of the known part Fb.

In this way, in the flow shown in FIG. 69, when the known part Ub of the nodal displacement is decided, the unknown part Fv, the unknown part Uv, and the known part Fb are calculated automatically. For a degree of freedom of a nodal external force for which a value is not designated, zero is set. In this way, as the degree of freedom satisfies the self-adjoint boundary condition, the analysis can be performed by the conventional-type finite element method, based on the set nodal displacement and nodal external force (S108).

(Exemplary Boundary Condition Setting 1 Applicable to Case of Non-Self-Adjoint Condition as Well)

FIG. 70 is a flowchart showing exemplary processing for setting a nodal displacement vector and a nodal external force vector that is also applicable to the case where a boundary condition is a non-self-adjoint condition, in an analysis in which a finite element is used. In the example shown in FIG. 70, the boundary condition decision unit 22 of the information processing device receives input of a degree of freedom of a nodal displacement and a value thereof (S201), and decides and records a degree of freedom of a known part Ub of a nodal displacement and a value thereof (S202). When the known part Ub is decided, an unknown part Fv of the nodal external force (S203), an unknown part Uv of the nodal displacement (S204) and a known part Fb (S205) are calculated automatically. At S205, for a degree of freedom of a nodal external force for which a value is not designated, zero is set.

The boundary condition decision unit 22 receives input of a degree of freedom of a nodal external force and a value thereof (S206), and corrects the degree of freedom and value of the nodal external force Fb to the input degree of freedom and value (S207). Further, the boundary condition decision unit 22 also receives input of an unknown part of the nodal displacement (S208). In the present example, a degree of freedom of a node for which a displacement is unknown is input. For example, the configuration may be such that a user can input the same by designating a node and a direction on a screen, or may be such that data indicating a degree of freedom of an unknown part are read in. The boundary condition decision unit 22 corrects the degree of freedom of the unknown part Uv, based on the input information (S209).

Similarly, the boundary condition decision unit 22 receives input of an unknown part of the nodal external force (S210), and corrects the degree of freedom of the unknown part Fv, based on the input information (S211). At S208 to S211, such a designation that designates both of the displacement and the external force as unknown parts with respect to the same degree of freedom can be received. In other words, the boundary condition decision unit 22 is capable of setting such a condition that different variables are set with respect to the same degree of freedom.

The determination unit 33 determines whether or not the degree of freedom satisfies the self-adjoint boundary condition (S212), and displays the determination result (S213). The determination unit 33 can determine, for example, whether or not the number of degrees of freedom of the known part Ub of the nodal displacement and the number of the unknown part Fv of the nodal external force are equal to each other. In the case where they are equal, it is determined that the self-adjoint boundary condition is satisfied, and in the case where they are not equal to each other, it is determined that the self-adjoint boundary condition is not satisfied.

For example, a boundary condition in which a plurality of variables (in the present example, the nodal displacement and the nodal external force) are known with respect to one direction of a certain node, that is, one degree of freedom, or a boundary condition in which a plurality of variables are unknown with respect to one degree of freedom, is considered not to satisfy a self-adjoint boundary condition. In the case where a boundary condition that does not satisfy a self-adjoint boundary condition is set, conventionally, it is impossible to execute analysis, but the use of the technique disclosed in the present specification (for example, the technique using the new-type finite element method) makes it possible to execute analysis under a non-self-adjoint boundary condition (S215). The analysis under a self-adjoint boundary condition is feasible by the conventional-type finite element method (S214).

It should be noted that the analysis processing at S214 or S215 might be executed by another device. For example, in the case where it is determined at S212 that the self-adjoint boundary condition is satisfied, the determination unit 33 execute the analysis processing at S214 by the conventional-type finite element method, and in the case where it is determined that the self-adjoint boundary condition is not satisfied, the determination unit 33 outputs information at S212 so as to enable the analysis processing at S215. The determination unit 33 can output information about, for example, how to obtain software for executing the processing at S215, guidance, etc.

(Exemplary Boundary Condition Setting 2 Applicable to Case of Non-Self-Adjoint Condition as Well)

FIG. 71 is a flowchart showing another exemplary processing for setting a nodal displacement vector and a nodal external force vector that is also applicable to the case where a boundary condition is a non-self-adjoint condition. In the example shown in FIG. 71, input of degrees of freedom and values of a nodal displacement and a nodal external force is received, and a degree of freedom that are not input particularly is set as an unknown part. The boundary condition decision unit 22 receives input of a degree of freedom of a nodal displacement and a value thereof (S301), fixes a degree of freedom of a known part Ub of the nodal displacement and a value thereof based on the input (S302), and with these, sets a degree of freedom of an unknown part Uv thereof as well (S303). Similarly, the boundary condition decision unit 22 receives input of a degree of freedom of the nodal external force and a value thereof (S304), fixes a degree of freedom of a known part Fb of the nodal external force and a value thereof based on the input (S305), and with this, fixes a degree of freedom of an unknown part Fv thereof as well (S306). Determination processing (S307), display processing (S308), and analysis processing (S309, 310) can be executed in the same manner as that in the case shown in FIG. 70.

In the example shown in FIG. 71, degrees of freedom of a nodal displacement and a nodal external force that are not input particularly are set as unknown parts. A user does not have to designate an unknown part clearly, but has to input and clearly set a value of zero for, for example, a node for which a displacement is to be set to zero. In contrast, in the example shown in FIG. 70, a variable of a degree of freedom for which a user has not designate as an unknown part or a known part is set to a certain value as a known part. Therefore, it is unnecessary for a user to set, regarding every degree of freedom, whether it is unknown or known. It should be noted that the information processing device might have a configuration such that a user is allowed to select which processing the user uses for inputting, either the processing shown in FIG. 70 or the processing shown in FIG. 71. It should be noted that the exemplary processing examples shown in FIGS. 70 and 71 are equivalent to the two processing methods (1) and (2) described in Section 11.7.1, respectively.

(Exemplary Analysis of Non-Self-Adjoint Problem with Respect to Finite Element Method)

FIG. 72 is a flowchart showing an exemplary analysis of a non-self-adjoint problem where a finite element method is used. In the example shown in FIG. 72, first of all, the information processing device 10 c inputs a calculation model (S401). For example, a finite element model, a shape, a material, an element, nodes, etc. of a structure to be processed are input.

The initial equation decision unit 21, using the data input at S401, creates element stiffness matrices, and creates a global stiffness matrix by superimposing the element stiffness matrices (S402). The global stiffness matrix created herein is exemplary initial equations expressed by a matrix. At S402, for example, two n-dimensional variable vectors each of which represents a physical amount at a node in a system in which the number of degrees of freedom is n, and an n-row stiffness matrix. More specifically, equations like eq. (441) or eq. (449) can be created.

The boundary condition decision unit 22 inputs data of a gravity load (S403), and creates a gravity load (S405). At S403, for example, data indicating a value of a gravity applied to each degree of freedom specified by a node and a direction, or a value of displacement thereof. Further, the boundary condition decision unit 22 inputs data that designate an unknown part of the nodal displacement and an unknown part of the nodal external force (S404). With the data input at S403 and S404, the known part and the unknown part of the nodal displacement, and the known part and the unknown part of the nodal external force are created (S405). It should be noted that the processing at S403, S404, and S405 can be executed in the same manner as, for example, that shown in FIG. 70.

The calculation unit 23 creates a matrix for use in an operation by transforming a stiffness matrix (S406). For example, based on two variable vectors having a degree of freedom of n, the calculation unit 23 generates a 2n-dimensional vector, and transforms the n-row matrix into a 2n-column matrix according to variables of the 2n-dimensional vector. For example, the matrix can be transformed into the form like eq. (450). The calculation unit 23 decides a known part that includes variables that become known by a boundary condition and an unknown part that includes unknown variables, among the variables of the 2n-dimensional vector. The 2n-column matrix and the 2n-dimensional vector are transformed so as to take formats such that the variables of the unknown part are expressed with the variables of the known part. For example, by using a transformed matrix as in eq. (452), variables of the unknown part can be calculated.

The calculation unit 23 calculates a solution of the equation of the matrix created at S406 (S407). In the case where only one solution exists as a result of the calculation, the calculation unit 23 creates a particular solution part of the nodal displacement and a particular solution part of the nodal external force (S408). For example, as in eq. (936), vectors that represent the particular solution of an unknown part can be created. In the case where a plurality of solutions exist, the calculation unit 23 defines a mode coefficient with respect to homogeneous solutions (S409). For example, let the number of homogeneous solutions be no, then, it is possible to define the no-dimensional vector {a_(o)} as a mode coefficient. The calculation unit 23 creates a homogeneous solution part of the nodal displacement and a homogeneous solution part of the nodal external force (S410). For example, as shown in eq. (937), a matrix that represent the homogeneous solution of an unknown part can be created.

At S411, the calculation unit 23 decides the solution of the unknown part of the nodal displacement by using, for example, eq. (938), and decides the solution of the unknown part of the nodal external force by using, for example, eq. (939). In the case where the solution can be decided uniquely as a result of the processing at S411, the result is output, and the analysis processing is ended (S412).

In the case where the solution cannot be decided uniquely, an optimal value of {a_(o)} is calculated by the least squares method. For example, the calculation unit 23 creates a matrix as shown in eq. (960) or eq. (961) (S413). Further, the calculation unit 23 decides a mode coefficient by, for example, solving eq. (962) regarding the mode coefficient {a_(o)} (S414). The calculation unit 23 outputs a solution corresponding to the mode coefficient in a predetermined range with the decided mode coefficient as the central value (S415). For example, the calculation unit 23 receives input of a value in the vicinities of the decided mode coefficient from a user, calculates a solution by using a mode coefficient of the input value, and outputs the calculated solution or information obtained from this solution. As the information obtained from the solution, for example, a transformed state, a stress distribution, an error distribution, or the like is output. Besides, a user interface may be provided so that the user can operate {a_(o)} with a slider, so as to designate a value in the predetermined range including the mode coefficient decided at S414.

It should be noted a program for executing the processing in the case where a plurality of solutions exist at S408 to S410, and a program for executing the processing at S401 to S407 may be provided separately. In other words, a program for executing at least one of the steps at S408 to S410 or a recording medium that records the program are encompassed in embodiments of the present invention.

[Exemplary Analysis Result]

FIG. 9 shows a result of determination about how peripheries should be supported in order to obtain, for a ring on which uniform gravity is acting, such a boundary condition about the inner edge thereof that “the displacement is zero and the surface force is also zero”. This problem has been considered insoluble conventionally, but by obtaining an analytical solution anew and comparing the same with a calculation result obtained by the eigenfunction method, it is confirmed that the calculation result is close to the analytical solution. From the figure, we can find that a large vertical stress or acts on an upper part and a lower part of the ring, and a shear stress T_(rθ) acts on left and right parts of the same, thereby being balanced with a gravity load. The inner edge is kept to be a perfect circle, from which we can find that no surface force acts. It should be noted that in FIG. 9, the left part shows the normal stress or, the center part shows the normal stress σ_(θ), and the right part shows shear the stress T_(rθ). The result shown in FIG. 9 is a result obtained by analytical calculation, without the ring being represented by finite elements. Exemplary specific primal simultaneous differential equations used in the analysis are shown below.

$\begin{matrix} \left\{ \begin{matrix} {{L_{1j}\varphi_{j}}\overset{\Delta}{=}{{{\left( {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right)\varphi_{1}} + {\mu \; \frac{\partial^{2}\varphi_{2}}{{\partial x}{\partial y}}}} = {\lambda^{*}\varphi_{1}^{*}}}} \\ {{L_{2j}\varphi_{j}}\overset{\Delta}{=}{{{\mu \; \frac{\partial^{2}\varphi_{1}}{{\partial x}{\partial y}}} + {\begin{pmatrix} {\frac{\partial^{2}}{\partial x^{2}} +} \\ {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}} \end{pmatrix}\varphi_{2}}} = {\lambda^{*}\varphi_{2}^{*}}}} \end{matrix} \right. & \left\lbrack {{Formula}\mspace{14mu} 270} \right\rbrack \\ {\left. \Rightarrow{L_{ij}\varphi_{j}} \right.\overset{\Delta}{=}{\lambda^{*}\varphi_{i}^{*}}} & \; \end{matrix}$

FIG. 10 shows an analysis result in the case of a square on which uniform gravity is acting, with a boundary condition for an upper edge thereof being such that “the displacement is zero and the surface force is also zero”. In other words, it is a result of solving the following problem: in order to make both of the displacement and the surface force of the upper edge of the square zero under uniform gravity, how should the other sides of the square be supported? In FIG. 10, the stress distributions are shown. The left part shows a normal stress σ_(x), the center part shows a normal stress σ_(y), and the right part shows a shear stress τ_(xy). In the left part of FIG. 10, we can see that large stresses σ_(x) and σ_(y) act on the lower edge of the square, whereby the displacement of the upper edge is kept to be zero. Here, the square is formulated as a finite element, and the element is only one. As a method of analysis with only one element, the p-method finite element is available. The analysis shown in FIG. 10 is an exemplary non-self-adjoint boundary condition, and an analysis with such a boundary condition cannot be performed by contemporary the p-method finite element method. Further, to which a boundary condition is set, either an edge or a node, depends on conveniences of formulation, and herein the formulation is such that a boundary condition is set with respect to a node.

FIG. 11 shows a result of analysis of in-plane deformation of a square plate having a fixed periphery. In FIG. 11, the stress distributions are shown. The left part shows a normal stress ox, the center part shows a normal stress σ_(y), the right part shows shear stress τ_(xy). In the center part of FIG. 11, we can see that large stresses σ_(y) act on the lower edge and the upper edge of the square. This indicates a self-adjoint boundary condition. The analysis result shown in FIG. 11 is identical to a result of analysis with respect to a p-method finite element. The analysis method of the above-described invention is considered to involve the p-method. In other words, by coding the finite element method based on the present invention, the p-method can be used seamlessly.

FIG. 12 shows a result of analysis of deformation by self-weight that occurs to a square plate element for out-of-plane deformation in the case where peripheries thereof are fixed. Here, the square is formulated as a finite element, and only one element is provided.

FIG. 13A shows a model of a beam, and FIG. 13B shows calculation results of primal eigenfunctions φ₁, φ₂, φ₃, and dual eigenfunctions φ₁*, φ₂*, φ₃* of the beam shown in FIG. 13A thereof having a left end under a free boundary condition and fixed boundary condition. FIG. 13C shows solutions (1-mode, 2-modes and 3-modes) determined by the eigenfunction method, and an analytical solution. The solution of 3-modes substantially overlaps the analytical solution.

FIG. 14 and FIG. 15 show analysis results in the case where a virtual boundary of a square is defined in a boundless region of the infinite, and a source (FIG. 14) or a vortex (FIG. 15) is arranged at the center thereof. The analysis results shown in FIGS. 14 and 15 show states of potential flow in the case where the Dirac's delta function is given to the right side of the above-described eq. (18) or (19) and calculation by the eigenfunction method is performed. We can confirm from these drawings that the state of the internal flow is close to the analytical solution.

[Exemplary Analysis] 10.1 Spring Damper System 10.1.1 Differential Equation

In the spring damper system shown in FIG. 16, let the damping coefficient be c, and let the spring constant be k. Let the displacement be u(t), and let the external force be f(t). The operator L is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 271} \right\rbrack & \; \\ {L \equiv {{c\frac{}{t}} + k}} & (456) \end{matrix}$

Then, we obtain a differential equation given as:

Lu(t)=f(t)  (457)

The constant γ is given as:

$\begin{matrix} {\gamma \equiv \frac{k}{c}} & (458) \end{matrix}$

This reciprocal number (1/γ) is called a time constant.

When the initial condition is given as

u(a)=u _(a),  (459)

We obtain an analytical solution of this as follows:

$\begin{matrix} {{u(t)} = {{\frac{1}{c}^{{- \gamma}\; t}{\int_{a}^{t}{{^{\gamma\xi} \cdot {f(\xi)}}\ {\xi}}}} + {u_{a}^{- {\gamma {({t - a})}}}}}} & (460) \end{matrix}$

Regarding an external force f(t), let a constant having dimensionality of force be F, and let a dimensionless function be p(t). Then, the external force is given as:

f(t)≡F·p(t)  (461)

Given a time range of a<t<b, dimensionless displacements are defined as:

$\begin{matrix} {{{\hat{u}(t)} \equiv {\frac{k}{F}{u(t)}}}{{{\hat{u}}_{a} \equiv {\frac{k}{F}u_{a}}},{{\hat{u}}_{b} \equiv {\frac{k}{F}u_{b}}}}} & (462) \end{matrix}$

Then, we obtain an analytical solution (460) as follows:

{circumflex over (u)}(t)=γe ^(−γt)∫_(a) ^(t) e ^(γξ) ·p(ξ)dξ+û _(a) e ^(−γ(t-a))  (463)

10.1.2 Adjoint Boundary Condition and Adjoint Differential Operators

[Formula 272]

According to eq. (58), multiplying eq. (457) by u* and integrating the same, we obtain:

∫_(a) ^(b) Lu·u*dt=∫ _(a) ^(b) f·u*dt  (464)

In the case of eq. (58), integration by region is performed, but in the present exemplary case, integration by time is performed. Partially integrating the left side of the equation, we obtain:

$\begin{matrix} {{\int_{a}^{b}{{{Lu} \cdot u^{*}}\ {t}}} = {\left\lbrack {cuu}^{*} \right\rbrack_{a}^{b} + {\int_{a}^{b}{{u \cdot \left( {{{- c}\frac{}{t}} + k} \right)}u^{*}\ {t}}}}} & (465) \end{matrix}$

The operator L* is given as:

$\begin{matrix} {L^{*} \equiv {{{- c}\frac{}{t}} + k}} & (466) \end{matrix}$

The boundary term R is given as:

R≡[cuu] _(a) ^(b)  (467)

Then, eq. (465) is transformed to:

∫_(a) ^(b) Lu·u*dt=R+∫ _(a) ^(b) u·L*u*dt  (468)

The operator satisfies:

L*≠L(469)

Therefore, this is a non-self-adjoint operator.

[Formula 273] [Exemplary Non-Self-Adjoint Boundary Condition 1]

Regarding the boundary term R, the initial condition is given as:

u(a)=0  (470)

Then, we obtain the following as an adjoint boundary condition:

u*(b)=0  (471)

As the conditions of eq. (470) and eq. (471) are different, this is a non-self-adjoint boundary condition. In the case of the initial condition of eq. (459), eq. (470) is given as a homogeneous boundary condition.

[Exemplary Non-Self-Adjoint Boundary Condition 2] Regarding the boundary term R, the initial condition is given as:

u(b)=0  (472)

Then, we obtain the following as an adjoint boundary condition:

u*(a)=0  (473)

As conditions of eq. (472) and eq. (473) are different, this is a non-self-adjoint boundary condition.

10.1.3 Homogenization of boundary condition

[Formula 274]

According to eq. (24), an index B is added to a term that satisfies an inhomogeneous boundary condition so as to let the term be u_(B), and an index H is added a term that satisfies a homogeneous boundary condition so as to let the term be u_(H). The displacement u is expressed as follows, with the sum of these:

u(t)≡u _(B)(t)+u _(H)(t)  (474)

In order that the boundary function u_(B) satisfies eq. (459) and is given as

u _(B)(a)=u _(a),  (475)

we define:

u _(B)(t)≡u _(a)  (476)

With this definition, the following is established, which is a useful property:

$\begin{matrix} {{Lu}_{B} = {{\left( {{c\frac{}{t}} + k} \right)u_{B}} = {ku}_{B}}} & (477) \end{matrix}$

Along with this, the following homogeneous boundary condition is imposed on

u _(H)(a)=0  (478)

This is equivalent to eq. (470).

When eq. (474) is substituted into eq. (457), the differential equation (457) is transformed to:

Lu _(H) =f _(H)  (479)

where

f _(H) ≡f−Lu _(E)  (480)

This is equivalent to eqs. (40), (41), and the problem is changed to a problem of determining a solution function u_(H) according to the boundary function u_(E).

10.1.4 Simultaneous Eigenvalue Problem and Eigenfunction Method [Formula 275]

In a simultaneous eigenvalue problem of

$\quad\begin{matrix} \left\{ {\begin{matrix} {{L\; \varphi} = {\lambda \; w\; \varphi^{*}}} \\ {{L^{*}\varphi^{*}} = {\lambda \; w\; \varphi}} \end{matrix},} \right. & (481) \end{matrix}$

the weight w is given as:

W≡k  (482)

The following boundary condition is imposed thereon:

$\quad\begin{matrix} \left\{ \begin{matrix} {{\varphi (a)} = 0} \\ {{\varphi^{*}(b)} = 0} \end{matrix} \right. & (483) \end{matrix}$

Then, this is equivalent to [Exemplary non-self-adjoint boundary condition 1] in Section 10.1.2, and the primal eigenfunction φ satisfies the condition of eq. (478). Then, according to eq. (69), let the solution function u_(H) be expressed as:

$\begin{matrix} {u_{H} \equiv {\sum\limits_{i}\; {c_{i}\varphi_{i}}}} & (484) \end{matrix}$

Substituting this into eq. (479), we obtain:

$\begin{matrix} {{{Lu}_{H} \equiv {\sum\limits_{i}\; {c_{i}L\; \varphi_{i}}}} = f_{H}} & (485) \end{matrix}$

Further, according to eq. (481), we obtain:

$\begin{matrix} {{\sum\limits_{i}\; {c_{i}\lambda_{i}w\; \varphi_{i}^{*}}} = f_{H}} & (486) \end{matrix}$

Multiplying both of the sides by φ_(j)* and integrating the same, we obtain:

$\begin{matrix} {{\sum\limits_{i}\; {c_{i}\lambda_{i}{\int_{a}^{b}{w\; {\varphi_{i}^{*} \cdot \varphi_{j}^{*}}\ {t}}}}} = {\int_{a}^{b}{{f_{H} \cdot \varphi_{j}^{*}}\ {t}}}} & (487) \end{matrix}$

According to the orthogonality, we obtain:

c _(i)λ_(i)∫_(a) ^(b) wφ _(i)*·φ_(i) *dt=∫ _(a) ^(b) f _(H)·φ_(i) *dt  (488)

In other words, the coefficient c_(i) decided to be:

$\begin{matrix} {c_{i} = \frac{\int_{a}^{b}{{f_{H} \cdot \varphi_{i}^{*}}\ {t}}}{\lambda_{i}{\int_{a}^{b}{w\; {\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}}}} & (489) \end{matrix}$

According to eqs. (461), (477), (480), and (482), this equation is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 276} \right\rbrack & \; \\ {c_{i} = {\frac{1}{\lambda_{i}}\left( {{\frac{F}{k}\frac{\int_{a}^{b}{{p \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}}} - \frac{\int_{a}^{b}{{u_{B} \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}}} \right)}} & (490) \end{matrix}$

Utilizing the weight w is a constant, eq. (489) is transformed, and the coefficient d_(i)* is defined as:

$\begin{matrix} {{d_{i}^{*} \equiv {c_{i}\lambda_{i}w}} = {\frac{\int_{a}^{b}{{f_{H} \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}} = {{F\frac{\int_{a}^{b}{{p \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}}} - {k\frac{\int_{a}^{b}{{u_{B} \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}}}}}} & (491) \end{matrix}$

Using this, eq. (486) is transformed to:

$\begin{matrix} {f_{H} = {\sum\limits_{i}{d_{i}^{*}\varphi_{i}^{*}}}} & (492) \end{matrix}$

This indicates that the external force term f_(H) is expanded by the dual eigenfunction φ_(i)*, and d_(i)* is an expansion coefficient thereof. The first term of the right side of the equation represents an expansion coefficient of an external force dimensionless function p(t), and the second term thereof represents an expansion coefficient of the boundary function u_(B)(t). Consequently, the solution displacement u(t) is given as:

$\begin{matrix} {{{u(t)} \equiv {{u_{B}(t)} + {u_{H}(t)}}}{{u_{B}(t)} \equiv u_{a}}{{u_{H}(t)} \equiv {\sum\limits_{i}{c_{i}{\varphi_{i}(t)}}}}{c_{i} \equiv {\frac{1}{\lambda_{i}}\left( {{\frac{F}{k}\frac{\int_{a}^{b}{{p \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}}} - \frac{\int_{a}^{b}{{u_{B} \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}}} \right)}}} & (493) \end{matrix}$

By making this dimensionless according to eq. (462), we obtain:

$\begin{matrix} {{{\hat{u}(t)} \equiv {{{\hat{u}}_{B}(t)} + {{\hat{u}}_{H}(t)}}}{{{\hat{u}}_{B}(t)} \equiv {\hat{u}}_{a}}{{{\hat{u}}_{H}(t)} \equiv {\sum\limits_{i}{{\hat{c}}_{i}{\varphi_{i}(t)}}}}{{\hat{c}}_{i} \equiv {\frac{1}{\lambda_{i}}\left( {\frac{\int_{a}^{b}{{p \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}} - \frac{\int_{a}^{b}{{{\hat{u}}_{B} \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}}} \right)}}} & (494) \end{matrix}$

10.1.5 Eigenfunction Sets [Formula 277]

From the simultaneous eigenvalue problem (481), the following equations on each eigenfunction are obtained:

L*Lφ=λ ² k ²φ  (495)

LL*φ*=λ ² k ²φ*  (496)

The operators satisfy:

$\begin{matrix} {{L^{*}L}\; = {{LL}^{*} = {{{- c^{2}}\frac{^{2}}{t^{2}}} + k^{2}}}} & (497) \end{matrix}$

Since eqs. (495) and (496) take the same form, eq. (495) may be solved so that the formats of φ,φ* can be obtained. Eq. (495) is given as:

$\begin{matrix} {{\left( {{{- c^{2}}\frac{^{2}}{t^{2}}} + k^{2}} \right)\varphi} = {\lambda^{2}k^{2}\varphi}} & (498) \end{matrix}$

And it is transformed to:

$\begin{matrix} {{\left( {\frac{^{2}}{t^{2}} - {\gamma^{2}\left( {1 - \lambda^{2}} \right)}} \right)\varphi} = 0} & (499) \end{matrix}$

This solution has a₁, a₂, a₁*, a₂* as arbitrary coefficients.

(1) When 0<λ<1, the following is given:

ω₁≡γ√{square root over (1−λ²)}  (500)

Then, we obtain:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {{a_{1}\sinh \; \omega_{1}t} + {a_{2}\cosh \; \omega_{1}t}}} \\ {\varphi^{*} = {{a_{1}^{*}\sinh \; \omega_{1}t} + {a_{2}^{*}\cosh \; \omega_{1}t}}} \end{matrix} \right. & (501) \end{matrix}$

(2) When λ=1, the following is given:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {{a_{1}t} + a_{2}}} \\ {\varphi^{*} = {{a_{1}^{*}t} + a_{2}^{*}}} \end{matrix} \right. & (502) \end{matrix}$

(3) When 1<λ, the following is given:

[Formula 278]

ω₃≡γ√{square root over (λ²−1)}  (503)

Then, we obtain:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {{a_{1}\sin \; \omega_{3}t} + {a_{2}\cos \; \omega_{3}t}}} \\ {\varphi^{*} = {{a_{1}^{*}\sin \; \omega_{3}t} + {a_{2}^{*}\cos \; \omega_{3}t}}} \end{matrix} \right. & (504) \end{matrix}$

Transforming the simultaneous eigenvalue problem (481), we obtain:

$\begin{matrix} \left\{ \begin{matrix} {{\left( {\frac{}{t} + \gamma} \right)\varphi} = {\lambda \; \gamma \; \varphi^{*}}} \\ {{\left( {\frac{}{t} - \gamma} \right)\varphi^{*}} = {{- \lambda}\; \gamma \; \varphi}} \end{matrix} \right. & (505) \end{matrix}$

The coefficients a₁, a₂, a₁*, a₂* can be decided so as to satisfy eq. (505). As a result of decision of the coefficients, the combinations we obtain are as follows.

(1) When 0<λ<1, the following two combinations are obtained:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {{\cosh \; \omega_{1}t} - {\frac{\omega_{1}}{\gamma}\sinh \; \omega_{1}t}}} \\ {\varphi^{*} = {\lambda \; \cosh \; \omega_{1}t}} \end{matrix} \right. & (506) \\ \left\{ \begin{matrix} {\varphi = {{\sinh \; \omega_{1}t} - {\frac{\omega_{1}}{\gamma}\cosh \; \omega_{1}t}}} \\ {\varphi^{*} = {{\lambda sinh\omega}_{1}t}} \end{matrix} \right. & (507) \end{matrix}$

(2) When λ=1, the following two combinations are obtained:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = 1} \\ {\varphi^{*} = 1} \end{matrix} \right. & (508) \\ \left\{ \begin{matrix} {\varphi = {{\gamma \; t} - 1}} \\ {\varphi^{*} = {\gamma \; t}} \end{matrix} \right. & (509) \end{matrix}$

(3) When 1<λ, the following two combinations are obtained:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {{\cos \; \omega_{3}t} + {\frac{\omega_{3}}{\gamma}\sin \; \omega_{3}t}}} \\ {\varphi^{*} = {\lambda \; \cos \; \omega_{3}t}} \end{matrix} \right. & (510) \\ \left\{ \begin{matrix} {\varphi = {{\sin \; \omega_{3}t} - {\frac{\omega_{3}}{\gamma}\cos \; \omega_{3}t}}} \\ {\varphi^{*} = {{\lambda sin\omega}_{3}t}} \end{matrix} \right. & (511) \end{matrix}$

These combinations only satisfy the simultaneous differential equations (505), and such conditions that these combinations should satisfy the boundary condition have not been imposed.

10.1.6 Eigenfunction [Formula 279]

In the previous section, three function combinations depending on the value of λ are shown. In the present section, combinations that satisfy the boundary condition eq. (483) are shown as follows:

(1) When 0<λ<1, no combination that satisfies the boundary condition (483) is available.

(2) The case where λ=¹ is not taken into consideration at the present stage, though it might have to be considered in a special case.

(3) When 1<λ, with respect to λ that satisfies the following characteristic equation (512), the following equations (513) are eigenfunctions:

$\begin{matrix} {{{\omega_{3}{\cos \left\lbrack {\left( {b - a} \right)\omega_{3}} \right\rbrack}} + {\gamma \; {\sin \left\lbrack {\left( {b - a} \right)\omega_{3}} \right\rbrack}}} = 0} & (512) \\ \left\{ \begin{matrix} {\varphi = {{- \lambda}\; {{\cos \left\lbrack {\omega_{3}\left( {b - a} \right)} \right\rbrack} \cdot {\sin \left\lbrack {\omega_{3}\left( {t - a} \right)} \right\rbrack}}}} \\ {\varphi^{*} = {\sin \left\lbrack {\omega_{3}\left( {b - t} \right)} \right\rbrack}} \end{matrix} \right. & (513) \end{matrix}$

A dimensionless quantity h is defined as:

h≡γ(b−a)  (514)

This results in that a termination time b is set as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 280} \right\rbrack & \; \\ {b = {a + \frac{h}{\gamma}}} & (515) \end{matrix}$

Further, a dimensionless time ξ is defined as:

$\begin{matrix} {\xi \equiv \frac{t - a}{b - a}} & (516) \end{matrix}$

This is transformed to:

t=a+ξ(b−a)  (517)

It should be noted that the dimensionless time η is defined as:

η≡γt

η_(a) ≡γa

η_(b) ≡γb  (518)

The following relationship exists between the two dimensionless times ξ,η:

$\begin{matrix} {t = {\frac{\eta}{\gamma} = {a + {\xi \left( {b - a} \right)}}}} & (519) \end{matrix}$

Therefore, they are associated by

$\begin{matrix} {{\xi = \frac{{a\; \gamma} - \eta}{\left( {a - b} \right)\gamma}}{and}} & (520) \\ {\eta = {\gamma {\left\{ {a + {\xi \left( {b - a} \right)}} \right\}.}}} & (521) \end{matrix}$

[Formula 281]

When these relationships are substituted into eqs. (512) and (513), the characteristic equations are transformed to functions of h,λ, and the eigenfunctions are transformed to functions of h,λ,ξ.

Let the dimensionless quantity h be:

h=15  (522)

Then, we obtain the first four eigenvalues as follows:

$\begin{matrix} {\lambda = \left\{ \begin{matrix} 1.0191 \\ 1.0748 \\ 1.1624 \\ 1.2763 \end{matrix} \right.} & (523) \end{matrix}$

Each eigenfunction is normalized so that the maximum value is 1, and is shown in FIG. 17.

10.1.7 Exemplary Analysis 1 [Formula 282]

The dimensionless function p(t) of eq. (461) is given as:

p(t)≡sin γt  (524)

We obtain the following analytical solution according to eq. (463):

{circumflex over (u)}(η)=½{(cos η_(a)−sin η_(a))e ^(−(η-η) ^(a) ⁾−(cos η−sin η)}û _(a) e ^(−(η-η) ^(a) ⁾  (525)

Let the initial condition be:

û _(a)=1 at η_(a)=0  (526)

Comparison between the analytical solutions of eq. (525) and the results obtained by the eigenfunction method of eq. (494) is shown in FIG. 18. In the drawing, the broken lines indicate the analytical solutions, and the solid lines indicate the results obtained by the eigenfunction method. “50M” means that the calculation is performed by using up to 50 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement û obtained by the calculation using about 10 modes by the eigenfunction method coincides well with the analytical solution.

10.1.8 Exemplary Analysis 2 [Formula 283]

The dimensionless function p(t) of eq. (461) is given as:

p(t)≡cos γt  (527)

According to eq. (463), the analytical solution is transformed to:

{circumflex over (u)}(η)=½{(cos η_(a)+sin η_(a))e ^(−(η-η) ^(a) ⁾+(cos η+sin η)}û _(a) e ^(−(η-η) ^(a) ⁾  (528)

Let the initial condition be:

û _(a)=1 at η_(a)=0  (529)

Comparison between the analytical solutions of eq. (528) and the results obtained by the eigenfunction method of eq. (494) is shown in FIG. 19. In the drawing, the broken lines indicate the analytical solutions, and the solid lines indicate the results obtained by the eigenfunction method. “50M” means that the calculation is performed by using up to 50 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement û obtained by the calculation using about 10 modes by the eigenfunction method coincides well with the analytical solution.

10.1.9 Exemplary Analysis 3 [Formula 284]

The dimensionless function p(t) of eq. (461) is given as:

p(t)≡e ^(−γt)  (530)

According to eq. (463), the analytical solution is transformed to:

{circumflex over (u)}(η)=(η−η_(a))e ^(−η) +û _(a) e ^(−(η-η) ^(a) ⁾  (531)

Let the initial condition be:

û _(a)=1 at η_(a)=0  (532)

Comparison between the analytical solutions of eq. (531), and the results obtained by the eigenfunction method of eq. (494) is shown in FIG. 20. In the drawing, the broken lines indicate the analytical solutions, and the solid lines indicate the results obtained by the eigenfunction method. “50M” means that the calculation is performed by using up to 50 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement û obtained by the calculation using about 10 modes by the eigenfunction method coincides well with the analytical solution.

10.2 Mass-Point Spring System 10.2.1 Differential Equation

Regarding a mass-point spring system shown in FIG. 21, let the mass of the point mass be m, and let the spring constant be k. Let the displacement and the external force be u(t) and f(t), respectively. The operator L is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 285} \right\rbrack & \; \\ {L \equiv {{m\frac{^{2}}{t^{2}}} + k}} & (533) \end{matrix}$

Then, we obtain the differential equation as follows:

Lu(t)=f(t)  (534)

Let the natural angular frequency ω_(o) be:

$\begin{matrix} {\omega_{o} \equiv \sqrt{\frac{k}{m}}} & (535) \end{matrix}$

When the initial condition is given as:

u(0)=u _(o),

u&(0)=v _(o)  (536)

we obtain an analytical solution thereof as follows:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 286} \right\rbrack} & \; \\ {{u(t)} = {{\frac{1}{m\; \omega_{o}}{\int_{0}^{t}{{{f(\xi)} \cdot {\sin\left\lbrack {\omega_{o}\left( {t - \xi} \right)}\  \right\rbrack}}{\xi}}}} + {u_{o}\cos \; \omega_{o}t} + {v_{o}\frac{1}{\omega_{o}}\sin \; \omega_{o}t}}} & (537) \end{matrix}$

Regarding the external force f(t), let the constant having dimensionality of force be F, and let the dimensionless function be p(t). Then, the following is given:

f(t)≡F·p(t)  (538)

Considering the time range as a<t<b, defining the dimensionless displacement as

$\begin{matrix} {{{\hat{u}(t)} \equiv {\frac{k}{F}{u(t)}}}{{{\hat{u}}_{a} \equiv {\frac{k}{F\;}u_{a}}},{{\hat{u}}_{b} \equiv {\frac{k}{F\;}v_{b}}},{{\hat{u}}_{o} \equiv {\frac{k}{F\;}u_{o}}},}} & (539) \end{matrix}$

and defining the dimensionless velocity as

$\begin{matrix} {{{\hat{v}}_{a} \equiv {\frac{k}{F\; \omega_{o}}v_{a}}},{{\hat{v}}_{b} \equiv {\frac{k}{F\; \omega_{o}}v_{b}}},{{\hat{v}}_{o} \equiv {\frac{k}{F\; \omega_{o}}v_{o}}},} & (540) \end{matrix}$

the analytical solution (537) is given as:

$\begin{matrix} {{\hat{u}(t)} = {{\omega_{o}{\int_{0}^{t}{{{p(\xi)} \cdot {\sin\left\lbrack {\omega_{o}\left( {t - \xi} \right)}\  \right\rbrack}}{\xi}}}} + {{\hat{u}}_{o}\cos \; \omega_{o}t} + {{\hat{v}}_{o}\sin \; \omega_{o}t}}} & (541) \end{matrix}$

10.2.2 Adjoint Boundary Condition and Adjoint Differential Operators [Formula 287]

Multiplying eq. (534) by u* and integrating the same according to eq. (58), we obtain:

∫_(a) ^(b) Lu·u*dt=∫ _(a) ^(b) f·u*dt  (542)

Whereas integration by region is performed with respect to eq. (58), this shows an example of integration by time. Performing partial integration with respect to the left side of the equation, we obtain:

$\begin{matrix} {\mspace{79mu} {{{\int_{a}^{b}{L\; {u \cdot u^{*}}\ {t}}} = {\left\lbrack {{m\text{?}} - {{mu}\text{?}}} \right\rbrack_{a}^{b} + {\int_{a}^{b}{{u \cdot \left( {m\frac{^{2}}{t^{2}}k} \right)}u^{*}\ {t}}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (543) \end{matrix}$

Let the operator L* be:

$\begin{matrix} {L^{*} \equiv {{m\frac{^{2}}{t^{2}}} + k}} & (544) \end{matrix}$

Let the boundary term R be:

R≡[mu&*−muu&*]_(a) ^(b)  (545)

Then, eq. (543) is transformed to:

∫_(a) ^(b) Lu·u*dt=R+∫ _(a) ^(b) u·L*u*dt  (546)

The operator satisfies:

L ^(*) =L,  (547)

Therefore, this is a self-adjoint operator.

[Formula 288] [Exemplary Non-Self-Adjoint Boundary Condition 1]

Regarding the boundary term R, the initial conditions are given as:

u(a)=0  (548)

Then, we obtain the following as adjoint boundary conditions:

u*(b)=0

u&*(b)=0  (549)

As the conditions of eq. (548) and eq. (549) are different, this is a non-self-adjoint boundary condition. In the case of the initial conditions of eq. (536), eq. (548) is given as a homogeneous boundary condition when a=0.

[Formula 289] [Exemplary Self-Adjoint Boundary Condition 1]

Regarding the boundary term R, the conditions of displacement are given as:

u(a)=0

u(b)=0  (550)

Then, we obtain the following as adjoint boundary conditions:

u*(a)=0

u*(b)=0  (551)

As the conditions of eq. (550) and eq. (551) coincide, this is a self-adjoint boundary condition.

10.2.3 Homogenization of Boundary Condition [Formula 290]

According to eq. (24), an index B is added to a term that satisfies an inhomogeneous boundary condition so as to let the term be u_(B), and an index H is added a term that satisfies a homogeneous boundary condition so as to let the term be u_(H). The displacement u is expressed as follows, with the sum of these:

(t)≡u _(B)(t)+u _(H)(t)  (552)

[Exemplary Non-Self-Adjoint Boundary Condition 1]

In the case where the initial conditions are

u(a)=u _(a)

u&(a)=v _(a)  (553)

we define the boundary function u_(B) as

u _(B)(t)≡u _(a) +v _(a)(t−a).  (554)

From this, we obtain:

u _(B)(a)=u _(a)

u&_(B)(a)=v _(a)  (555)

Therefore, the following homogeneous boundary conditions are imposed on u:

u _(H)(a)=0

u&_(H)(a)=0  (556)

This is equivalent to eq. (548). Thus, this example becomes a non-self-adjoint problem. It should be noted that the following is established, which is a useful property:

$\begin{matrix} {{L\; u_{B}} = {{\left( {{c\frac{}{t}} + k} \right)u_{B}} = {ku}_{B}}} & (557) \end{matrix}$

[Formula 291]

[Exemplary Self-Adjoint Boundary Condition 1] In the case where the displacement conditions are

u(a)=u _(a)

u(b)=u _(b)  (558)

we define the boundary function u_(B) as

$\begin{matrix} {{u_{B}(t)} \equiv {u_{a} + {\frac{u_{b} - u_{a}}{b - a}{\left( {t - a} \right).}}}} & (559) \end{matrix}$

From this, we obtain:

u _(B)(a)=u _(a)

u _(B)(b)=u _(b)  (560)

Therefore, the following homogeneous boundary conditions are imposed on u_(H):

u _(H)(a)=0

u _(H)(b)=0  (561)

This is equivalent to eq. (550). Thus, this example becomes a self-adjoint problem. It should be noted that the following is established, which is a useful property:

$\begin{matrix} {{L\; u_{B}} = {{\left( {{c\frac{}{t}} + k} \right)u_{B}} = {ku}_{B}}} & (562) \end{matrix}$

When eq. (552) is substituted into eq. (534), the differential equation (534) is transformed to:

Lu _(H) =f _(H)  (563)

where

f _(H) ≡f−Lu _(B)  (564)

This is equivalent to eqs. (40), (41), and the problem is changed to a problem of determining a solution function ti according to the boundary function u_(B).

10.2.4 Simultaneous Eigenvalue Problem and Eigenfunction Method 1

In the case of [Exemplary non-self-adjoint boundary condition 1], this is a method for solving a non-self-adjoint problem in which the boundary function u_(B) of eq. (554) is used. In a simultaneous eigenvalue problem of

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 292} \right\rbrack & \; \\ \left\{ {\begin{matrix} {{L\; \varphi} = {\lambda \; w\; \varphi^{*}}} \\ {{L^{*}\varphi^{*}} = {\lambda \; w\; \varphi}} \end{matrix},} \right. & (565) \end{matrix}$

the weight w is given as:

W≡k  (566)

The following boundary condition is imposed thereon:

$\begin{matrix} {\mspace{79mu} \left\{ {\begin{matrix} {{\varphi (a)} = 0} \\ {{\text{?}(a)} = 0} \\ {{\varphi^{*}(b)} = 0} \\ {{\text{?}(b)} = 0} \end{matrix}\text{?}\text{indicates text missing or illegible when filed}} \right.} & (567) \end{matrix}$

Then, the primal eigenfunction φ satisfies the condition of eq. (548). Then, according to eq. (69), let the solution function u be expressed as:

$\begin{matrix} {u_{H} \equiv {\sum\limits_{i}{c_{i}\; \varphi_{i}}}} & (568) \end{matrix}$

Substituting this into eq. (563), we obtain:

$\begin{matrix} {{{Lu}_{H} \equiv {\sum\limits_{i}{c_{i}L\; \varphi_{i}}}} = f_{H}} & (569) \end{matrix}$

Further, according to eq. (565), we obtain:

$\begin{matrix} {{\sum\limits_{i}{c_{i}\lambda_{i}w\; \varphi_{i}^{*}}} = f_{H}} & (570) \end{matrix}$

Multiplying both of the sides by φ_(j)* and integrating the same, we obtain:

$\begin{matrix} {{\sum\limits_{i}{c_{i}\lambda_{i}{\int_{a}^{b}{w\; {\varphi_{i}^{*} \cdot \varphi_{j}^{*}}\ {t}}}}} = {\int_{a}^{b}{{f_{H} \cdot \varphi_{j}^{*}}\ {t}}}} & (571) \end{matrix}$

According to the orthogonality, we obtain:

c _(i)λ_(i)∫_(a) ^(b) wφ _(i)*·φ_(i) *dt=∫ _(a) ^(b) f _(H)·φ_(i) *dt  (572)

In other words, the coefficient c_(i) is decided to be:

$\begin{matrix} {c_{i} = \frac{\int_{a}^{b}{{f_{H} \cdot \varphi_{i}^{*}}\ {t}}}{\lambda_{i}{\int_{a}^{b}{w\; {\varphi_{i}^{*}\  \cdot \varphi_{i}^{*}}{t}}}}} & (573) \end{matrix}$

Then, it is transformed to:

$\begin{matrix} {c_{i} = {\frac{1}{\lambda_{i}}\left( {{\frac{F}{k}\frac{\int_{a}^{b}{{p \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}\; {{\varphi_{i}^{*}\  \cdot \varphi_{i}^{*}}{t}}}} - \frac{\int_{a}^{b}{{u_{B} \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*}\  \cdot \varphi_{i}^{*}}{t}}}} \right)}} & (574) \end{matrix}$

[Formula 293]

Utilizing the weight w is a constant, eq. (573) is transformed, and the coefficient d_(i)* is defined as:

$\begin{matrix} \begin{matrix} {{d_{i}^{*} \equiv {c_{i}\lambda_{i}w}} = \frac{\int_{a}^{b}{{f_{H} \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}}} \\ {= {{F\frac{\int_{a}^{b}{{p \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}}} - {k\frac{\int_{a}^{b}{{u_{B} \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {t}}}}}} \end{matrix} & (575) \end{matrix}$

Using this, eq. (570) is transformed to:

$\begin{matrix} {f_{H} = {\sum\limits_{i}{d_{i}^{*}\varphi_{i}^{*}}}} & (576) \end{matrix}$

This indicates that the external force term f_(H) is expanded by the dual eigenfunction φ_(i)*, and d_(i)* is an expansion coefficient thereof. The first term of the right side of the equation represents an expansion coefficient of an external force dimensionless function p(t), and the second term thereof represents an expansion coefficient of the boundary function u_(B)(t).

Consequently, the solution displacement u(t) is given as:

$\begin{matrix} {{{u(t)} \equiv {{u_{B}(t)} + {u_{H}(t)}}}{{u_{B}(t)} \equiv {u_{a} + {v_{a}\left( {t - a} \right)}}}{{u_{H}(t)} \equiv {\sum\limits_{i}{c_{i}{\varphi_{i}(t)}}}}{c_{i} \equiv {\frac{1}{\lambda_{i}}\left( {{\frac{F}{k}\frac{\int_{a}^{b}{{p \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}\; {{\varphi_{i}^{*}\  \cdot \varphi_{i}^{*}}{t}}}} - \frac{\int_{a}^{b}{{u_{B} \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*}\  \cdot \varphi_{i}^{*}}{t}}}} \right)}}} & (577) \end{matrix}$

Making this dimensionless gives:

$\begin{matrix} {{{\hat{u}(t)} \equiv {{{\hat{u}}_{B}(t)} + {{\hat{u}}_{H}(t)}}}{{{\hat{u}}_{B}(t)} \equiv {{\hat{u}}_{a} + {{\hat{v}}_{a}{\omega_{o}\left( {t - a} \right)}}}}{{{\hat{u}}_{H}(t)} \equiv {\sum\limits_{i}{{\hat{c}}_{i}{\varphi_{i}(t)}}}}{{\hat{c}}_{i} \equiv {\frac{1}{\lambda_{i}}\left( {\frac{\int_{a}^{b}{{p \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}\; {{\varphi_{i}^{*}\  \cdot \varphi_{i}^{*}}{t}}} - \frac{\int_{a}^{b}{{{\hat{u}}_{B} \cdot \varphi_{i}^{*}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}^{*}\  \cdot \varphi_{i}^{*}}{t}}}} \right)}}} & (578) \end{matrix}$

10.2.5 Simultaneous Eigenvalue Problem and Eigenfunction Method 2 [Formula 291]

In the case of [Exemplary self-adjoint boundary condition 1], this is simplified as a self-adjoint problem, without use of a method for solving a non-self-adjoint problem. If the boundary conditions of

$\begin{matrix} \left\{ \begin{matrix} {{\varphi (a)} = 0} \\ {{\varphi (b)} = 0} \\ {{\varphi^{*}(a)} = 0} \\ {{\varphi^{*}(b)} = 0} \end{matrix} \right. & (579) \end{matrix}$

are imposed, the primal eigenfunction φ satisfies conditions of eq. (550).

In the simultaneous eigenvalue problem of

$\begin{matrix} \left\{ {\begin{matrix} {{L\; \varphi} = {\lambda \; w\; \varphi^{*}}} \\ {{L^{*}\varphi^{*}} = {\lambda \; w\; \varphi}} \end{matrix},} \right. & (580) \end{matrix}$

φ and φ* are equivalent. Therefore, the form they take is either

φ*=φ

Lφ=λwφ,  (581)

or

φ*=−φ

Lφ=−λwφ.  (582)

In the case of this problem, the form of eq. (582) is taken.

In the case of eq. (582), in the procedure up to when eqs. (577), and (578) are obtained, φ* may be changed to (−φ), and the boundary function u_(B) of eq. (559) may be used. As a result, the solution displacement u(t) is given as:

$\begin{matrix} {{{{u(t)} \equiv {{u_{B}(t)} + {u_{H}(t)}}}{u_{B}(t)} \equiv {u_{a} + {\frac{u_{b} - u_{a}}{b - a}\left( {t - a} \right)}}}{{u_{H}(t)} \equiv {\sum\limits_{i}{c_{i}{\varphi_{i}(t)}}}}{c_{i} \equiv {{- \frac{1}{\lambda_{i}}}\left( {{\frac{F}{k}\frac{\int_{a}^{b}{{p \cdot \varphi_{i}}\ {t}}}{\int_{a}^{b}\; {{\varphi_{i}\  \cdot \varphi_{i}}{t}}}} - \frac{\int_{a}^{b}{{u_{B} \cdot \varphi_{i}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}\  \cdot \varphi_{i}}{t}}}} \right)}}} & (583) \end{matrix}$

Making this dimensionless gives:

$\begin{matrix} {{{\hat{u}(t)} \equiv {{{\hat{u}}_{b}(t)} + {{\hat{u}}_{H}(t)}}}{{{\hat{u}}_{B}(t)} \equiv {{\hat{u}}_{a} + {\frac{{\hat{u}}_{b} - {\hat{u}}_{a}}{b - a}\left( {t - a} \right)}}}{{{\hat{u}}_{H}(t)} \equiv {\sum\limits_{i}{{\hat{c}}_{i}{\varphi_{i}(t)}}}}{{\hat{c}}_{i} \equiv {{- \frac{1}{\lambda_{i}}}\left( {\frac{\int_{a}^{b}{{p \cdot \varphi_{i}}\ {t}}}{\int_{a}^{b}\; {{\varphi_{i}\  \cdot \varphi_{i}}{t}}} - \frac{\int_{a}^{b}{{{\hat{u}}_{B} \cdot \varphi_{i}}\ {t}}}{\int_{a}^{b}{{\varphi_{i}\  \cdot \varphi_{i}}{t}}}} \right)}}} & (584) \end{matrix}$

10.2.6 Eigenfunction Sets [Formula 295]

From the simultaneous eigenvalue problem (565), we obtain the following equations on each eigenfunction:

L*Lφ=λ ² k ²φ  (585)

LL*φ*=λ ² k ²φ*  (586)

The operators satisfy:

$\begin{matrix} {{L^{*}L} = {{LL}^{*} = {{m^{2}\frac{^{4}}{t^{4}}} + {2{mk}\frac{^{2}}{t^{2}}} + k^{2}}}} & (587) \end{matrix}$

Since eqs. (585) and (586) take the same form, eq. (585) may be solved to obtain a format with 4) and V. Eq. (585) is as follows:

$\begin{matrix} {{\left( {{m^{2}\frac{^{4}}{t^{4}}} + {2{mk}\frac{^{2}}{t^{2}}} + k^{2}} \right)\varphi} = {\lambda^{2}k^{2}\varphi}} & (588) \end{matrix}$

This is transformed to:

$\begin{matrix} {{\left( {\frac{^{2}}{t^{2}} + {\omega_{o}^{2}\left( {1 - \lambda} \right)}} \right)\left( {\frac{^{2}}{t^{2}} + {\omega_{o}^{2}\left( {1 + \lambda} \right)}} \right)\varphi} = 0} & (589) \end{matrix}$

This solution has a₁, a₂, a₃, a₄, a₁*, a₂*, a₃*, a₄* as arbitrary coefficients.

(1) When 0<λ<1, the following is given:

ω₁≡ω_(o)√{square root over (1−λ)}

ω₂≡ω_(o)√{square root over (1+λ)}  (590)

[Formula 296]

Then, we obtain:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {{a_{1}\sin \; \omega_{1}t} + {a_{2}\cos \; \omega_{1}t} + {a_{3}\sin \; \omega_{2}t} + {a_{4}\cos \; \omega_{2}t}}} \\ {\varphi^{*} = {{a_{1}^{*}\sin \; \omega_{1}t} + {a_{2}^{*}\cos \; \omega_{1}t} + {a_{3}^{*}\sin \; \omega_{2}t} + {a_{4}^{*}\cos \; \omega_{2}t}}} \end{matrix} \right. & (591) \end{matrix}$

(2) When λ=1, the following is given:

ω₄≡ω_(o)√{square root over (2)}  (592)

Then, we obtain:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {{a_{1}t} + a_{2} + {a_{3}\sin \; \omega_{4}t} + {a_{4}\cos \; \omega_{4}t}}} \\ {\varphi^{*} = {{a_{1}^{*}t} + a_{2}^{*} + {a_{3}^{*}\sin \; \omega_{4}t} + {a_{4}^{*}\cos \; \omega_{4}t}}} \end{matrix} \right. & (593) \end{matrix}$

(3) When 1<A, the following is given:

ω₅≡ω_(o)√{square root over (λ−1)}

ω₆≡ω_(o)√{square root over (1−λ)}  (594)

Then, we obtain:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {{a_{1}\sinh \; \omega_{5}t} + {a_{2}\cosh \; \omega_{5}t} + {a_{3}\sin \; \omega_{6}t} + {a_{4}\cos \; \omega_{6}t}}} \\ {\varphi^{*} = {{a_{1}^{*}\sinh \; \omega_{5}t} + {a_{2}^{*}\cosh \; \omega_{5}t} + {a_{3}^{*}\sin \; \omega_{6}t} + {a_{4}^{*}\cos \; \omega_{6}t}}} \end{matrix} \right. & (595) \end{matrix}$

Transforming the simultaneous eigenvalue problem (565), we obtain:

$\begin{matrix} \left\{ \begin{matrix} {{\left( {\frac{^{2}}{t^{2}} + \omega_{o}^{2}} \right)\varphi} = {\lambda \; \omega_{0}^{2}\varphi^{*}}} \\ {{\left( {\frac{^{2}}{t^{2}} + \omega_{o}^{2}} \right)\varphi^{*}} = {\lambda \; \omega_{0}^{2}\varphi}} \end{matrix} \right. & (596) \end{matrix}$

The coefficients a₁, a₂, a₃, a₄, a₁*, a₂*, a₃*, a₄* can be decided so as to satisfy eq. (596). As a result of decision of the coefficients, the combinations we obtain are as follows.

[Formula 297]

(1) When 0<λ<1, the following four combinations are obtained:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {\cos \; \omega_{1}t}} \\ {\varphi^{*} = {\cos \; \omega_{1}t}} \end{matrix} \right. & (597) \\ \left\{ \begin{matrix} {\varphi = {\sin \; \omega_{1}t}} \\ {\varphi^{*} = {\sin \; \omega_{1}t}} \end{matrix} \right. & (598) \\ \left\{ \begin{matrix} {\varphi = {{- \cos}\; \omega_{2}t}} \\ {\varphi^{*} = {\cos \; \omega_{2}t}} \end{matrix} \right. & (599) \\ \left\{ \begin{matrix} {\varphi = {{- \sin}\; \omega_{2}t}} \\ {\varphi^{*} = {\sin \; \omega_{2}t}} \end{matrix} \right. & (600) \end{matrix}$

(2) When λ=1, the following four combinations are obtained:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = 1} \\ {\varphi^{*} = 1} \end{matrix} \right. & (601) \\ \left\{ \begin{matrix} {\varphi = t} \\ {\varphi^{*} = t} \end{matrix} \right. & (602) \\ \left\{ \begin{matrix} {\varphi = {{- \cos}\; \omega_{4}t}} \\ {\varphi^{*} = {\cos \; \omega_{4}t}} \end{matrix} \right. & (603) \\ \left\{ \begin{matrix} {\varphi = {{- \sin}\; \omega_{4}t}} \\ {\varphi^{*} = {\sin \; \omega_{4}t}} \end{matrix} \right. & (604) \end{matrix}$

(3) When 1<λ, the following four combinations are obtained:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {\cosh \; \omega_{5}t}} \\ {\varphi^{*} = {\cosh \; \omega_{5}t}} \end{matrix} \right. & (605) \\ \left\{ \begin{matrix} {\varphi = {\sinh \; \omega_{5}t}} \\ {\varphi^{*} = {\sinh \; \omega_{5}t}} \end{matrix} \right. & (606) \\ \left\{ \begin{matrix} {\varphi = {{- \cos}\; \omega_{6}t}} \\ {\varphi^{*} = {\cos \; \omega_{6}t}} \end{matrix} \right. & (607) \\ \left\{ \begin{matrix} {\varphi = {{- \sin}\; \omega_{6}t}} \\ {\varphi^{*} = {\sin \; \omega_{6}t}} \end{matrix} \right. & (608) \end{matrix}$

These combinations only satisfy the simultaneous differential equations (596), and such conditions that these combinations should satisfy the boundary condition have not been imposed.

10.2.7 Eigenfunction [Formula 298]

In the previous section, three function combinations depending on the value of λ are shown. In the present section, combinations that satisfy the boundary condition eq. (567) are shown as follows.

(1) When 0<λ<1, with respect to λ that satisfies the following characteristic equation (609), the following equations (610) are eigenfunctions:

$\begin{matrix} {{{2\; \omega_{1}\omega_{2}} + {2\; \omega_{1}\omega_{2}{\cos \left\lbrack {\left( {b - a} \right)\omega_{1}} \right\rbrack}{\cos \left\lbrack {\left( {b - a} \right)\omega_{2}} \right\rbrack}} + {\left( {\omega_{1}^{2} + \omega_{2}^{2}} \right){\sin \left\lbrack {\left( {b - a} \right)\omega_{1}} \right\rbrack}{\sin \left\lbrack {\left( {b - a} \right)\omega_{2}} \right\rbrack}}} = 0} & (609) \\ \left\{ \begin{matrix} {\varphi = {{\omega_{1}\begin{Bmatrix} {{{- {\cos \left\lbrack {\omega_{1}\left( {t - a} \right)} \right\rbrack}} \cdot {\sin \left\lbrack {\omega_{2}\left( {b - a} \right)} \right\rbrack}} + {\sin \left\lbrack {\omega_{2}\left( {b - t} \right)} \right\rbrack} -} \\ {{\cos \left\lbrack {\omega_{1}\left( {b - a} \right)} \right\rbrack} \cdot {\sin \left\lbrack {\omega_{2}\left( {t - a} \right)} \right\rbrack}} \end{Bmatrix}} +}} \\ {\omega_{2}\begin{Bmatrix} {{{\cos \left\lbrack {\omega_{2}\left( {b - a} \right)} \right\rbrack} \cdot {\sin \left\lbrack {\omega_{1}\left( {t - a} \right)} \right\rbrack}} - {\sin \left\lbrack {\omega_{1}\left( {b - t} \right)} \right\rbrack} +} \\ {{\cos \left\lbrack {\omega_{2}\left( {t - a} \right)} \right\rbrack} \cdot {\sin \left\lbrack {\omega_{1}\left( {b - a} \right)} \right\rbrack}} \end{Bmatrix}} \\ \begin{matrix} {\varphi^{*} = {{\omega_{1}\begin{Bmatrix} {{{- {\cos \left\lbrack {\omega_{1}\left( {t - a} \right)} \right\rbrack}} \cdot {\sin \left\lbrack {\omega_{2}\left( {b - a} \right)} \right\rbrack}} - {\sin \left\lbrack {\omega_{2}\left( {b - t} \right)} \right\rbrack} +} \\ {{\cos \left\lbrack {\omega_{1}\left( {b - a} \right)} \right\rbrack} \cdot {\sin \left\lbrack {\omega_{2}\left( {t - a} \right)} \right\rbrack}} \end{Bmatrix}} +}} \\ {\omega_{2}\begin{Bmatrix} {{{\cos \left\lbrack {\omega_{2}\left( {b - a} \right)} \right\rbrack} \cdot {\sin \left\lbrack {\omega_{1}\left( {t - a} \right)} \right\rbrack}} - {\sin \left\lbrack {\omega_{1}\left( {b - t} \right)} \right\rbrack} -} \\ {{\cos \left\lbrack {\omega_{2}\left( {t - a} \right)} \right\rbrack} \cdot {\sin \left\lbrack {\omega_{1}\left( {b - a} \right)} \right\rbrack}} \end{Bmatrix}} \end{matrix} \end{matrix} \right. & (610) \end{matrix}$

(2) The case where λ=1 is not taken into consideration at the present stage, though it might have to be considered in a special case.

[Formula 299]

(3) When 1<λ, with respect to λ that satisfies the following characteristic equation (611), the following equations (612) are eigenfunctions:

$\begin{matrix} {{{2\; \omega_{5}\omega_{6}} + {2\; \omega_{5}\omega_{6}{\cosh \left\lbrack {\left( {b - a} \right)\omega_{5}} \right\rbrack}{\cos \left\lbrack {\left( {b - a} \right)\omega_{6}} \right\rbrack}} + {\left( {{- \omega_{5}^{2}} + \omega_{6}^{2}} \right){\sinh \left\lbrack {\left( {b - a} \right)\omega_{5}} \right\rbrack}{\sin \left\lbrack {\left( {b - a} \right)\omega_{6}} \right\rbrack}}} = 0} & (611) \\ \left\{ \begin{matrix} {\varphi = {{\omega_{5}\begin{Bmatrix} {{{- {\cosh \left\lbrack {\omega_{5}\left( {t - a} \right)} \right\rbrack}} \cdot {\sin \left\lbrack {\omega_{6}\left( {b - a} \right)} \right\rbrack}} + {\sin \left\lbrack {\omega_{6}\left( {b - t} \right)} \right\rbrack} -} \\ {{\cosh \left\lbrack {\omega_{5}\left( {b - a} \right)} \right\rbrack} \cdot {\sin \left\lbrack {\omega_{6}\left( {t - a} \right)} \right\rbrack}} \end{Bmatrix}} -}} \\ {\omega_{6}\begin{Bmatrix} {{{\cos \left\lbrack {\omega_{6}\left( {b - a} \right)} \right\rbrack} \cdot {\sinh \left\lbrack {\omega_{5}\left( {t - a} \right)} \right\rbrack}} - {\sinh \left\lbrack {\omega_{5}\left( {b - t} \right)} \right\rbrack} +} \\ {{\cos \left\lbrack {\omega_{6}\left( {t - a} \right)} \right\rbrack} \cdot {\sinh \left\lbrack {\omega_{5}\left( {b - a} \right)} \right\rbrack}} \end{Bmatrix}} \\ \begin{matrix} {\varphi^{*} = {{\omega_{5}\begin{Bmatrix} {{{- {\cosh \left\lbrack {\omega_{5}\left( {t - a} \right)} \right\rbrack}} \cdot {\sin \left\lbrack {\omega_{6}\left( {b - a} \right)} \right\rbrack}} - {\sin \left\lbrack {\omega_{6}\left( {b - t} \right)} \right\rbrack} +} \\ {{\cosh \left\lbrack {\omega_{5}\left( {b - a} \right)} \right\rbrack} \cdot {\sin \left\lbrack {\omega_{6}\left( {t - a} \right)} \right\rbrack}} \end{Bmatrix}} +}} \\ {\omega_{6}\begin{Bmatrix} {{{\cos \left\lbrack {\omega_{6}\left( {b - a} \right)} \right\rbrack} \cdot {\sinh \left\lbrack {\omega_{5}\left( {t - a} \right)} \right\rbrack}} - {\sinh \left\lbrack {\omega_{5}\left( {b - t} \right)} \right\rbrack} -} \\ {{\cos \left\lbrack {\omega_{6}\left( {t - a} \right)} \right\rbrack} \cdot {\sinh \left\lbrack {\omega_{5}\left( {b - a} \right)} \right\rbrack}} \end{Bmatrix}} \end{matrix} \end{matrix} \right. & (612) \end{matrix}$

A dimensionless quantity h is defined as:

h≡ω _(o)(b−a)  (613)

This results in that a termination time b is set as follows:

$\begin{matrix} {b = {a + \frac{h}{\omega_{o}}}} & (614) \end{matrix}$

Further, a dimensionless time ξ is defined as:

$\begin{matrix} {\xi \equiv \frac{t - a}{b - a}} & (615) \end{matrix}$

This is transformed to:

t=a+ξ(b−a)  (616)

It should be noted that the dimensionless time η is defined as:

η≡ω_(o) t

η_(a)≡ω_(o) a

η_(b)≡ω_(o) b  (617)

The following relationship exists between the two dimensionless times ξ,η:

[Formula 300]

$\begin{matrix} {t = {\frac{\eta}{\omega_{o}} = {a + {\xi \left( {b - a} \right)}}}} & (618) \end{matrix}$

Therefore, they are associated by

$\begin{matrix} {{\xi = \frac{{a\; \omega_{o}} - \eta}{\left( {a - b} \right)\omega_{o}}}{and}} & (619) \\ {\eta = {\omega_{o}{\left\{ {a + {\xi \left( {b - a} \right)}} \right\}.}}} & (620) \end{matrix}$

When these relationships are substituted into eqs. (609) and (611), the characteristic equations are transformed to functions of h,λ. When these relationships are substituted into eqs. (610) and (612), the eigenfunctions are transformed to functions of h,λ,ξ.

Let the dimensionless quantity h be:

h=15  (621)

Then, we obtain the first two eigenvalues of eq. (609) as follows:

$\begin{matrix} {\lambda = \left\{ \begin{matrix} 0.19894 \\ 0.21840 \end{matrix} \right.} & (622) \end{matrix}$

Each eigenfunction in eq. (610) is normalized so that the maximum value is 1, and is shown in FIG. 22.

Likewise, the first two eigenvalues of eq. (611) are:

[Formula 301]

$\begin{matrix} {\lambda = \left\{ \begin{matrix} 1.0776 \\ 1.6098 \end{matrix} \right.} & (623) \end{matrix}$

Each eigenfunction in eq. (612) is normalized so that the maximum value is 1, and is shown in FIG. 23.

10.2.8 Exemplary analysis1

[Formula 302]

The dimensionless function p(t) of eq. (538) is given as:

p(t)≡sin ω_(o) t  (624)

We obtain the following analytical solution according to eq. (541):

{circumflex over (u)}(η)=½(−η cos η+sin η)+{circumflex over (u)} cos η+{circumflex over (v)} _(o) sin η  (625)

Let the initial condition be:

û _(a)=5, {circumflex over (v)}_(a)=0 at η_(a)=0  (626)

Comparison between the analytical solutions of eq. (625) and the results obtained by the eigenfunction method of eq. (578) is shown in FIG. 24.

In the drawing, the broken lines indicate the analytical solutions, and the solid lines indicate the results obtained by the eigenfunction method. “56M” means that the calculation is performed by using up to 56 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement û obtained by the calculation using about 10 modes by the eigenfunction method coincides well with the analytical solution.

10.2.9 Self-Adjoint Eigenfunction [Formula 303]

As the problem is a self-adjoint problem under the boundary condition of eq. (558), an eigenfunction that satisfies eq. (579) is used.

To summarize this, the differential equation satisfies

$\begin{matrix} {{{\left( {\frac{^{2}}{t^{2}} + {\omega_{o}^{2}\left( {1 + \lambda} \right)}} \right)\varphi} = 0},} & (627) \end{matrix}$

and as the boundary condition, the following is imposed:

$\begin{matrix} \left\{ \begin{matrix} {{\varphi (a)} = 0} \\ {{\varphi (b)} = 0} \end{matrix} \right. & (628) \end{matrix}$

In this case, the n-th eigenvalue λ_(n) is given as:

$\begin{matrix} {\lambda_{n} = {\left\{ \frac{n\; \pi}{\left( {b - a} \right)\omega_{o}} \right\}^{2} - 1}} & (629) \end{matrix}$

The n-th eigenfunction φ_(n) is given as:

$\begin{matrix} {\varphi_{n} = {\sin \left\lbrack {\frac{n\; \pi}{b - a}\left( {t - a} \right)} \right\rbrack}} & (630) \end{matrix}$

10.2.10 Exemplary Analysis 2 [Formula 304]

The dimensionless function p(t) of eq. (538) is given as:

p(t)≡sin ω_(o) t  (631)

According to eq. (541), we obtain an analytical solution given as:

$\begin{matrix} {{\hat{u}(\eta)} = {{\frac{1}{2}\left( {{{- \eta}\; \cos \; \eta} + {\eta_{b}\frac{\cos \; \eta_{b}}{\sin \; \eta_{b\;}}\sin \; \eta}} \right)} + {{\hat{u}}_{o}\left( {{\cos \; \eta} - {\frac{\cos \; \eta_{b}}{\sin \; \eta_{b}}\sin \; \eta}} \right)} + {{\hat{u}}_{b}\frac{\sin \; \eta}{\sin \; \eta_{b}}}}} & (632) \end{matrix}$

Let the condition of displacement be:

û _(a)=5, û _(b)=0, η_(a)=0  (633)

Comparison between the analytical solutions of eq. (632) and the results obtained by the eigenfunction method of eq. (584) is shown in FIG. 25. In the drawing, the broken lines indicate the analytical solutions, and the solid lines indicate the results obtained by the eigenfunction method. “50M” means that the calculation is performed by using up to 50 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement ti obtained by the calculation using about 10 modes by the eigenfunction method coincides well with the analytical solution.

10.2.11 Hamilton's Principle [Formula 305]

Regarding the mass-point spring system, we have discussed that the very differential operators are self-adjoint, but the boundary condition is non-self-adjoint, in some cases. We found that in such a case, it is necessary to solve the problem by using eigenfunctions of a non-self-adjoint problem. Here, the time t is taken as an integration variable of eq. (122), and Exemplary analysis 1 shows that the problem can be solved by:

∫_(a) ^(b) Lu·δu*dt=∫ _(a) ^(b) f·δu*dt  (634)

Special exemplary cases of this equation include a self-adjoint problem. Exemplary analysis 2 shows that the problem can be solved as a self-adjoint problem by:

∫_(a) ^(b) Lu·δudt=∫ _(a) ^(b) f·δudt  (635)

This is nothing but the Hamilton's principle. In other words, this shows that the present method involves not only the principle of virtual work but also the Hamilton's principle.

[Formula 306]

Generally, according to the Hamilton's principle, the both sides of the differential equation (534) are multiplied by δu and are integrated, whereby eq. (635) is obtained, and thereafter, by partial integration, the left side of the equation is transformed to:

$\begin{matrix} {{\int_{a}^{b\;}{{{Lu} \cdot \delta}\; u{t}}} = {{{\int_{a}^{b}{{\left( {m\; \frac{^{2}}{t^{2}}u} \right) \cdot \delta}\; u\; {t}}} + {\int_{a}^{b}{{({ku}) \cdot \delta}\; u{t}}}} = {{\left\lbrack {{\left( {m\; \frac{}{t}u} \right) \cdot \delta}\; u} \right\rbrack_{a}^{b} - {\int_{a}^{b}{{\left( {m\frac{}{t}u} \right) \cdot \left( {\frac{}{t}\delta \; u} \right)}{t}}} + {\int_{a}^{b}{{\delta \left( {\frac{1}{2}{ku}^{2}} \right)}{t}}}} = {\left\lbrack {{\left( {m\; \frac{}{t}u} \right) \cdot \delta}\; u} \right\rbrack_{a}^{b} - {\int_{a}^{b}{\delta \left\{ {\frac{1}{2}{m\left( {\frac{}{t}u} \right)}^{2}} \right\} {t}}} + {\int_{a}^{b}{{\delta \left( {\frac{1}{2}{ku}^{2}} \right)}{t}}}}}}} & (636) \end{matrix}$

Here, a condition such that variations are zero at both end points of time, which is as follows, is imposed:

$\begin{matrix} \left\{ \begin{matrix} {{\delta \; {u(a)}} = 0} \\ {{\delta \; {u(b)}} = 0} \end{matrix} \right. & (637) \end{matrix}$

Thereby, eq. (636) is transformed to:

$\begin{matrix} {{\int_{a}^{b}{{{Lu} \cdot \delta}\; u{t}}} = {\int_{a}^{b}{\delta \left\{ {{{- \frac{1}{2}}{m\left( {\frac{}{t}u} \right)}^{2}} + {\frac{1}{2}{ku}^{2}}} \right\} {t}}}} & (638) \end{matrix}$

Here, we should note that the condition of eq. (637) is the boundary condition of the self-adjoint eigenfunction.

Kinetic energy T is given as:

$\begin{matrix} {T \equiv {\frac{1}{2}{m\left( {\frac{}{t}u} \right)}^{2}}} & (639) \end{matrix}$

[Formula 307]

Potential energy U is given as:

U≡½ku ²  (640)

The Lagrangian function P is given as:

P≡U−T  (641)

Then, eq. (638) is transformed to:

∫_(a) ^(b) Lu·δudt=∫ _(a) ^(b) δPdt  (642)

For example, in free vibration, the external force term f is zero. Therefore, by eliminating the symbol δ of variation from the integration, we obtain the following transformed equation:

δ∫_(a) ^(b) Pdt=0  (643)

This is called the Hamilton's principle, which is the variational principle in the self-adjoint problem.

10.3 Static Deflection of String

10.3.1 Differential equation

In static deflection of a string shown in FIG. 26, let the length of the string be l, and let the tensile force be T. Let the displacement be u(x), and let the distributed load be f(x). Then, the operator L is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 308} \right\rbrack & \; \\ {L \equiv {{- T}\frac{^{2}}{x^{2}}}} & (644) \end{matrix}$

Then, we obtain the following differential equation:

Lu(x)=f(x)  (645)

When boundary conditions are given as

u(0)=u _(o)

u&(0)=θo,  (646)

we obtain the following analytical solution:

$\begin{matrix} {{u(x)} = {{{- \frac{1}{T}}{\int_{0}^{x}{\int_{0}^{\xi}{{f(\xi)}{\xi}{\xi}}}}} + u_{o} + {\theta_{o}x}}} & (647) \end{matrix}$

When boundary conditions are given as:

u(0)=u _(o)

u(λ)=u _(λ),  (648)

we obtain the following analytical solution:

$\begin{matrix} {{u(x)} = {{\frac{1}{T}\left( {{\frac{x}{\lambda}{\int_{0}^{\lambda}{\int_{0}^{\xi}{{f(\xi)}{\xi}{\xi}}}}} - {\int_{0}^{x}{\int_{0}^{\xi}{{f(\xi)}{\xi}{\xi}}}}} \right)} + {\left( {1 - \frac{x}{\lambda}} \right)u_{o}} + {\frac{x}{\lambda}u_{\lambda}}}} & (649) \end{matrix}$

[Formula 309]

Regarding a distributed load f(x), let a constant having dimensionality of (force/length) be Q, and let a dimensionless function be p(x). Then, the distributed load is given as:

f(x)≡Q·p(x)  (650)

Since terms having dimensionality of force are T and Ql,

$\begin{matrix} \frac{Q\; \lambda}{T} & (651) \end{matrix}$

is dimensionless. A dimensionless position ξ is given as:

$\begin{matrix} {\xi \equiv \frac{x}{\lambda}} & (652) \end{matrix}$

Dimensionless displacements are defined as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 310} \right\rbrack & \; \\ {{{\hat{u}(x)} \equiv {\frac{T}{Q\; \lambda^{2\;}}{u(x)}}}{{{\hat{u}}_{o} \equiv {\frac{T}{Q\; \lambda^{2}}u_{o}}},{{\hat{u}}_{\lambda} \equiv {\frac{T}{Q\; \lambda^{2}}u_{\lambda}}}}} & (653) \end{matrix}$

Dimensionless angles are defined as:

$\begin{matrix} {{{\hat{\theta}}_{o} \equiv {\frac{T}{Q\; \lambda}\theta_{o}}},{{\hat{\theta}}_{\lambda} \equiv {\frac{T}{Q\; \lambda}\theta_{\lambda}}}} & (654) \end{matrix}$

Then, we obtain an analytical solution (647) as follows:

$\begin{matrix} {{\hat{u}(x)} = {{{- \frac{1}{\lambda^{2}}}{\int_{0}^{x}{\int_{0}^{\xi}{{p(\xi)}{\xi}{\xi}}}}} + {\hat{u}}_{o} + {{\hat{\theta}}_{o}\frac{x}{\lambda}}}} & (655) \end{matrix}$

We obtain an analytical solution (649) as follows:

$\begin{matrix} {{\hat{u}(x)} = {{\frac{1}{\lambda^{2\;}}\left( {{\frac{x}{\lambda}{\int_{0}^{\lambda}{\int_{0}^{\xi}{{p(\xi)}{\xi}{\xi}}}}} - {\int_{0}^{x}{\int_{0}^{\xi}{{f(\xi)}{\xi}{\xi}}}}} \right)} + {\left( {1 - \frac{x}{\lambda}} \right){\hat{u}}_{o}} + {\frac{x}{\lambda}{\hat{u}}_{\lambda}}}} & (656) \end{matrix}$

10.3.2 Adjoint Boundary Condition and Adjoint Differential Operators [Formula 311]

Multiplying eq. (645) by u* and integrating the same, we obtain:

∫₀ ^(λ) Lu·u*dx=∫ ₀ ^(b) f·u*dx  (657)

By partial integration of the left side of the equation, we obtain:

$\begin{matrix} {{\int_{0}^{\lambda}{{{Lu} \cdot u^{*}}{x}}} = {{T\left\lbrack {{u{\overset{.}{u}}^{*}} - {\overset{.}{u}u^{*}}} \right\rbrack}_{0}^{\lambda} + {\int_{0}^{\lambda}{{u \cdot \left( {{- T}\frac{^{2}}{x^{2}}} \right)}u^{*}{x}}}}} & (658) \end{matrix}$

Let the operator L* be:

$\begin{matrix} {L^{*} \equiv {{- T}\frac{^{2}}{x^{2\;}}}} & (659) \end{matrix}$

Let the boundary term R be:

R≡T[uu&*−u&*]₀ ^(λ)  (660)

Then, eq. (658) is transformed to:

∫₀ ^(λ) Lu·u*dx=R+∫ ₀ ^(λ) u·L*u*dx  (661)

The operator satisfies:

L*=L,  (662)

Therefore, this is a self-adjoint operator.

[Formula 312] [Exemplary Non-Self-Adjoint Boundary Condition 1]

Regarding the boundary term R, the boundary conditions are given as:

u(0)=0

u&(0)=0  (663)

Then, we obtain the following as adjoint boundary conditions:

u*(λ)=0

u&*(λ)=0  (664)

As the conditions of eq. (663) and eq. (664) are different, this is a non-self-adjoint boundary condition.

[Exemplary Self-Adjoint Boundary Condition 1]

Regarding the boundary term R, the boundary conditions are given as:

u(0)=0

u(λ)=0  (665)

Then, we obtain the following as adjoint boundary conditions:

u*(0)=0

u*(λ)=0  (666)

As the conditions of eq. (665) and eq. (666) coincide, this is a self-adjoint boundary condition.

10.3.3 Homogenization of Boundary Condition

An index B is added to a term that satisfies an inhomogeneous boundary condition so as to let the term be u_(B), and an index H is added a term that satisfies a homogeneous boundary condition so as to let the term be u_(H). The displacement u is expressed as follows, with the sum of these:

[Formula 313]

u(x)≡u _(B)(x)+u _(H)(x)  (667)

[Exemplary Non-Self-Adjoint Boundary Condition 1]

In the case where the boundary conditions are

u(0)=u _(o)

u&(0)=θ_(o),  (668)

we define the boundary function u_(B) as

$\begin{matrix} {{u_{B}(x)} \equiv {{u_{o}\cos \; \frac{x}{\lambda}} + {\lambda \; \theta_{o}\sin \; {\frac{x}{\lambda}.}}}} & (669) \end{matrix}$

Accordingly, we obtain:

u _(B)(0)=u _(o)

u&_(B)(0)=θ_(o)  (670)

Therefore, the following homogeneous boundary conditions are imposed on u:

u _(H)(0)=0

u&_(H)(0)=0  (671)

This example becomes a non-self-adjoint problem.

It should be noted that the following is established, which is a useful property:

$\begin{matrix} {{Lu}_{B} = {{\left( {{- T}\; \frac{^{2}}{x^{2}}} \right)u_{B}} = {\frac{T}{\lambda^{2}}u_{B}}}} & (672) \end{matrix}$

[Formula 314] [Exemplary Self-Adjoint Boundary Condition 1]

In the case where the boundary conditions are

u(0)=u _(o)

u(λ)=u _(λ)  (673)

we define the boundary function u_(B) as

$\begin{matrix} {{u_{\; B}(x)} \equiv {\frac{1}{\sin \; 1}{\left\{ {{u_{o}\; {\sin \left( {1 - \frac{x}{\lambda}} \right)}} + {u_{\lambda}{\sin \left( \frac{x}{\lambda} \right)}}} \right\}.}}} & (674) \end{matrix}$

Accordingly, we obtain:

u _(B)(0)=u _(o)

u _(B)(λ)=u _(λ)  (675)

Therefore, the following homogeneous boundary conditions are imposed on

u _(H)(0)=0

u _(H)(λ)=0  (676)

This example becomes a self-adjoint problem.

It should be noted that the following is established, which is a useful property:

$\begin{matrix} {{Lu}_{B} = {{\left( {{- T}\frac{^{2}}{x^{2}}} \right)u_{B}} = {\frac{T}{\lambda^{2}}u_{B}}}} & (677) \end{matrix}$

When eq. (667) is substituted into eq. (645), the differential equation (645) is transformed to:

Lu _(H) =f _(H)  (678)

where

f _(H) ≡f−Lu _(B)  (679)

This is equivalent to eqs. (40), (41), and the problem is changed to a problem of determining a solution function u_(H) according to the boundary function u_(B).

10.3.4 Eigenfunction Sets [Formula 315]

In a simultaneous eigenvalue problem of

$\begin{matrix} \left\{ {\begin{matrix} {{L\; \varphi} = {\lambda \; w\; \varphi^{*}}} \\ {{L^{*}\varphi^{*}} = {\lambda \; w\; \varphi}} \end{matrix},} \right. & (680) \end{matrix}$

the weight w is given as:

$\begin{matrix} {w \equiv \frac{T}{\lambda^{2}}} & (681) \end{matrix}$

From the simultaneous eigenvalue problem, the following equations on each eigenfunction are obtained:

$\begin{matrix} {{L^{*}L\; \varphi} = {{\lambda^{2}\left( \frac{T}{\lambda^{2}} \right)}^{2}\varphi}} & (682) \\ \left\lbrack {{Formula}\mspace{14mu} 316} \right\rbrack & \; \\ {{{LL}^{*}\varphi^{*}} = {{\lambda^{2}\left( \frac{T}{\lambda^{2}} \right)}^{2}\varphi^{*}}} & (683) \end{matrix}$

The operators satisfy:

$\begin{matrix} {{L^{*}L} = {{LL}^{*} = {T^{2}\frac{^{4}}{x^{4}}}}} & (684) \end{matrix}$

Since eqs. (682) and (683) take the same form, eq. (682) may be solved so that the formats of φ,φ* can be obtained. Eq. (682) is given as:

$\begin{matrix} {{\left( {T^{2}\frac{^{4}}{x^{4}}} \right)\varphi} = {{\lambda^{2}\left( \frac{T}{\lambda^{2}} \right)}^{2}\varphi}} & (685) \end{matrix}$

And it is transformed to,

$\begin{matrix} {{\left( {\frac{^{2}}{x^{2}} - {\frac{1}{\lambda^{2}}\lambda}} \right)\left( {\frac{^{2}}{x^{2}} + {\frac{1}{\lambda^{2}}\lambda}} \right)\varphi} = 0} & (686) \end{matrix}$

This solution has a₁, a₂, a₃, a₄, a₁*, a₂*, a₃*, a₄* as arbitrary coefficients. Given

$\begin{matrix} {{\omega \equiv {\frac{1}{\lambda}\sqrt{\lambda}}},} & (687) \end{matrix}$

we obtain:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {{a_{1}\sin \; \omega \; x} + {a_{2}\cos \; \omega \; x} + {a_{3}\sinh \; \omega \; x} + {a_{4}\cosh \; \omega \; x}}} \\ {\varphi^{*} = {{a_{1}^{*}\sin \; \omega \; x} + {a_{2}^{*}\cos \; \omega \; x} + {a_{3}^{*}\sinh \; \omega \; x} + {a_{4}^{*}\cosh \; \omega \; x}}} \end{matrix} \right. & (688) \end{matrix}$

Transforming the simultaneous eigenvalue problem (680) gives:

$\begin{matrix} \left\{ \begin{matrix} {{\left( \frac{^{2}}{x^{2}} \right)\varphi} = {{- \lambda}\; \frac{1}{\lambda^{2}}\varphi^{*}}} \\ {{\left( \frac{^{2}}{x^{2}} \right)\varphi^{*}} = {{- \lambda}\; \frac{1}{\lambda^{2}}\varphi}} \end{matrix} \right. & (689) \end{matrix}$

The coefficient a₁, a₂, a₃, a₄, a₁*, a₂*, a₃*, a₄* can be decided so as to satisfy eq. (689). As a result of decision of the coefficients, we can obtain the following four combinations.

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {\cos \; \omega \; x}} \\ {\varphi^{*} = {\cos \; \omega \; x}} \end{matrix} \right. & (690) \\ \left\{ \begin{matrix} {\varphi = {\sin \; \omega \; x}} \\ {\varphi^{*} = {\sin \; \omega \; x}} \end{matrix} \right. & (691) \\ \left\{ \begin{matrix} {\varphi = {{- \cosh}\; \omega \; x}} \\ {\varphi^{*} = {\cosh \; \omega \; x}} \end{matrix} \right. & (692) \\ \left\{ \begin{matrix} {\varphi = {{- \sinh}\; \omega \; x}} \\ {\varphi^{*} = {\sinh \; \omega \; x}} \end{matrix} \right. & (693) \end{matrix}$

These combinations only satisfy the simultaneous differential equations (680), and such conditions that these combinations should satisfy the boundary condition have not been imposed.

10.3.5 Eigenfunction [Formula 317] [Exemplary Non-Self-Adjoint Boundary Condition 1]

In the case where the homogeneous boundary condition is expressed by eq. (671), the following conditions are imposed:

$\begin{matrix} \left\{ \begin{matrix} {{\varphi (0)} = 0} \\ {{\overset{.}{\varphi}(0)} = 0} \\ {{\varphi^{*}(\lambda)} = 0} \\ {{{\overset{.}{\varphi}}^{*}(\lambda)} = 0} \end{matrix} \right. & (694) \end{matrix}$

As a result, we obtain the following characteristic equation:

1+cos ωλ cos h ωλ=0  (695)

With respect to λ satisfying this, the following are eigenfunctions:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {{\left( {{\cos \; \omega \; \lambda} + {\cosh \; {\omega\lambda}}} \right)\left( {{\sin \; \omega \; x} - {\sinh \mspace{2mu} \omega \; x}} \right)} -}} \\ {\left( {{\sin \; {\omega\lambda}} + {\sinh \; {\omega\lambda}}} \right)\left( {{\cos \; \omega \; x} - {\cosh \; \omega \; x}} \right)} \\ {\varphi^{*} = {{\left( {{\cos \; \omega \; \lambda} + {\cosh \; {\omega\lambda}}} \right)\left( {{\sin \; \omega \; x} + {\sinh \; \omega \; x}} \right)} -}} \\ {\left( {{\sin \; \omega \; \lambda} + {\sinh \; \omega \; \lambda}} \right)\left( {{\cos \; \omega \; x} + {\cosh \; \omega \; x}} \right)} \end{matrix} \right. & (696) \end{matrix}$

When eqs. (652) and (687) are substituted therein, the characteristic equation becomes a function of λ, and the eigenfunctions become functions of λ,ξ. The first four eigenvalues are as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 318} \right\rbrack & \; \\ {\lambda = \left\{ \begin{matrix} 3.5160 \\ 22.034 \\ 61.697 \\ 120.90 \end{matrix} \right.} & (697) \end{matrix}$

Each eigenfunction is normalized so that the maximum value is 1, and is shown in FIG. 27.

[Formula 319] [Exemplary Self-Adjoint Boundary Condition 1]

In the case where the homogeneous boundary condition is expressed by eq. (676), the following conditions are imposed:

$\begin{matrix} \left\{ \begin{matrix} {{\varphi (0)} = 0} \\ {{\varphi (\lambda)} = 0} \\ {{\varphi^{*}(0)} = 0} \\ {{\varphi^{*}(\lambda)} = 0} \end{matrix} \right. & (698) \end{matrix}$

As a result, we obtain the n-th eigenvalue λ_(n) given as:

λ_(n)=(nπ)²  (699)

We also obtain the n-th eigenfunction φ_(n) given as:

$\begin{matrix} {\varphi_{n} = {\sin \left( {\frac{n\; \pi}{\lambda}x} \right)}} & (700) \end{matrix}$

10.3.6 Eigenfunction Method [Formula 320] [Exemplary Non-Self-Adjoint Boundary Condition 1]

A solution displacement u(x) is given as:

$\begin{matrix} {{{u(x)} \equiv {{u_{B}(x)} + {u_{H}(x)}}}{{u_{B}(x)} \equiv {{u_{o}\cos \; \frac{x}{\lambda}} + {\lambda \; \theta_{o}\sin \; \frac{x}{\lambda}}}}{{u_{H}(x)} \equiv {\sum\limits_{i}{c_{i}{\varphi_{i}(x)}}}}c_{i} \equiv {\frac{1}{\lambda_{i}}\left( {{\frac{Q\; \lambda^{2}}{T}\frac{\int_{0}^{\lambda}{{p \cdot \varphi_{i}^{*}}{x}}}{\int_{0}^{\lambda}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}{x}}}} - \frac{\int_{0}^{\lambda}{{u_{E} \cdot \varphi_{i}^{*}}{x}}}{\int_{0}^{\lambda}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}{x}}}} \right)}} & (701) \end{matrix}$

Making this dimensionless gives:

$\begin{matrix} {{{\hat{u}(x)} \equiv {{{\hat{u}}_{B}(x)} + {{\hat{u}}_{H}(x)}}}{{{\hat{u}}_{B}(x)} \equiv {{{\hat{u}}_{o}\cos \; \frac{x}{\lambda}} + {{\hat{\theta}}_{o}\sin \; \frac{x}{\lambda}}}}{{{\hat{u}}_{H}(x)} \equiv {\sum\limits_{i}{{\hat{c}}_{i}{\varphi_{i}(x)}}}}{{\hat{c}}_{i} \equiv {\frac{1}{\lambda_{i}}\left( {\frac{\int_{0}^{\lambda}{{p \cdot \varphi_{i}^{*}}{x}}}{\int_{0}^{\lambda}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}{x}}} - \frac{\int_{0}^{\lambda}{{{\hat{u}}_{B} \cdot \varphi_{i}^{*}}{x}}}{\int_{0}^{\lambda}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}{x}}}} \right)}}} & (702) \end{matrix}$

[Exemplary Self-Adjoint Boundary Condition 1]

A solution displacement u(x) is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 321} \right\rbrack & \; \\ {{{u(x)} \equiv {{u_{B}(x)} + {u_{H}(x)}}}{{u_{B}(x)} \equiv {\frac{1}{\sin \; 1}\left\{ {{u_{o}{\sin \left( {1 - \frac{x}{\lambda}} \right)}} + {u_{\lambda}\sin \; \left( \frac{x}{\lambda} \right)}} \right\}}}{{u_{H}(x)} \equiv {\sum\limits_{i}{c_{i}{\varphi_{i}(x)}}}}c_{i} \equiv {\frac{1}{\lambda_{i}}\left( {{\frac{Q\; \lambda^{2}}{T}\frac{\int_{0}^{\lambda}{{p \cdot \varphi_{i}}{x}}}{\int_{0}^{\lambda}{{\varphi_{i} \cdot \varphi_{i}}{x}}}} - \frac{\int_{0}^{\lambda}{{u_{B} \cdot \varphi_{i}}{x}}}{\int_{0}^{\lambda}{{\varphi_{i} \cdot \varphi_{i}}{x}}}} \right)}} & (703) \end{matrix}$

Making this dimensionless gives:

$\begin{matrix} {{{\hat{u}(x)} \equiv {{{\hat{u}}_{B}(x)} + {{\hat{u}}_{H}(x)}}}{{{\hat{u}}_{B}(x)} \equiv {\frac{1}{\sin \; 1}\left\{ {{{\hat{u}}_{o}{\sin \left( {1 - \frac{x}{\lambda}} \right)}} + {{\hat{u}}_{\lambda}{\sin \left( \frac{x}{\lambda} \right)}}} \right\}}}{{{\hat{u}}_{H}(x)} \equiv {\sum\limits_{i}{{\hat{c}}_{i}{\varphi_{i}(x)}}}}{{\hat{c}}_{i} \equiv {\frac{1}{\lambda_{i}}\left( {\frac{\int_{0}^{\lambda}{{p \cdot \varphi_{i}}{x}}}{\int_{0}^{\lambda}{{\varphi_{i} \cdot \varphi_{i}}{x}}} - \frac{\int_{0}^{\lambda}{{{\hat{u}}_{B} \cdot \varphi_{i}}{x}}}{\int_{0}^{\lambda}{{\varphi_{i} \cdot \varphi_{i}}{x}}}} \right)}}} & (704) \end{matrix}$

10.3.7 Exemplary Non-Self-Adjoint Analysis 1 [Formula 322]

The dimensionless function p(x) of eq. (650) is given as:

$\begin{matrix} {{p(x)} \equiv {\sin \; \frac{\pi}{\lambda}x}} & (705) \end{matrix}$

We obtain the following analytical solution according to eq. (655):

$\begin{matrix} {{\hat{u}(\xi)} = {\frac{{{- \pi}\; \xi} + {\sin \left( {\pi \; \xi} \right)}}{\pi^{2}} + {\hat{u}}_{o} + {{\hat{\theta}}_{o}\xi}}} & (706) \end{matrix}$

Let the boundary condition be:

û _(o)=0, {circumflex over (θ)}_(o)=0.1  (707)

Comparison between the analytical solutions of eq. (706) and the results obtained by the eigenfunction method of eq. (702) is shown in FIG. 28. In the drawing, the broken lines indicate the analytical solution and the black solid lines indicate the results obtained by the eigenfunction method. “30M” means that the calculation is performed by using up to 30 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement a obtained by the calculation using about 3 modes by the eigenfunction method coincides well with the analytical solution.

10.3.8 Exemplary Non-Self-Adjoint Analysis 2 [Formula 323]

The dimensionless function p(x) of eq. (650) is given as:

$\begin{matrix} {{p(x)} \equiv {\cos \; \frac{\pi}{\lambda}x}} & (708) \end{matrix}$

According to eq. (655), the analytical solution is transformed to:

$\begin{matrix} {{\hat{u}(\xi)} = {\frac{{- 1} + {\cos \left( {\pi \; \xi} \right)}}{\pi^{2}} + {\hat{u}}_{o} + {{\hat{\theta}}_{o}\xi}}} & (709) \end{matrix}$

Let the boundary condition be:

û _(o)=0, {circumflex over (θ)}_(o)=0.1  (710)

Comparison between the analytical solutions of eq. (709) and the results obtained by the eigenfunction method of eq. (702) is shown in FIG. 29. In the drawing, the broken lines indicate the analytical solutions, and the black solid lines indicate the results obtained by the eigenfunction method. “30M” means that the calculation is performed by using up to 30 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement û obtained by the calculation using about 3 modes by the eigenfunction method coincides well with the analytical solution.

10.3.9 Exemplary Self-Adjoint Analysis 1 [Formula 324]

The dimensionless function p(x) of eq. (650) is given as:

$\begin{matrix} {{p(x)} \equiv {\sin \; \frac{\pi}{\lambda}x}} & (711) \end{matrix}$

According to eq. (656), the analytical solution is transformed to:

$\begin{matrix} {{\hat{u}(\xi)} = {\frac{\sin \left( {\pi \; \xi} \right)}{\pi^{2}} + {\left( {1 - \xi} \right){\hat{u}}_{o}} + {{\hat{u}}_{\lambda}\xi}}} & (712) \end{matrix}$

Let the boundary condition be:

û _(o)=0, û_(λ)==0.1  (713)

Comparison between the analytical solutions of eq. (713) and the results obtained by the eigenfunction method of eq. (704) is shown in FIG. 30. In the drawing, the broken lines indicate the analytical solutions, and the black solid lines indicate the results obtained by the eigenfunction method. “30M” means that the calculation is performed by using up to 30 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement a obtained by the calculation using about 3 modes by the eigenfunction method coincides well with the analytical solution.

10.3.10 Exemplary Self-Adjoint Analysis 2 [Formula 325]

The dimensionless function p(x) of eq. (650) is given as:

$\begin{matrix} {{p(x)} \equiv {\cos \frac{\pi}{\lambda}x}} & (714) \end{matrix}$

According to eq. (656), the analytical solution is transformed to:

$\begin{matrix} {{\hat{u}(\xi)} = {\frac{{- 1} + {2\; \xi} + {\cos \left( {\pi \; \xi} \right)}}{\pi^{2}} + {\left( {1 - \xi} \right){\hat{u}}_{o}} + {{\hat{u}}_{\lambda}\xi}}} & (715) \end{matrix}$

Let the boundary condition be:

û _(o)=0, û _(λ)=0.1  (716)

Comparison between the analytical solutions of eq. (715) and the results obtained by the eigenfunction method of eq. (704) is shown in FIG. 31. In the drawing, the broken lines indicate the analytical solutions, and the black solid lines indicate the results obtained by the eigenfunction method. “30M” means that the calculation is performed by using up to 30 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement a obtained by the calculation using about 3 modes by the eigenfunction method coincides well with the analytical solution.

10.4 Static Deflection of Beam 10.4.1 Differential Equation

In static deflection of a beam shown in FIG. 32, let the length of the beam be l, and let the bending stiffness be EI. Let the shear force be F(x), and let the bending moment be M(x). Further, let the displacement be u(x), and let the distributed load be f(x). Then, the operator L is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 326} \right\rbrack & \; \\ {L \equiv {{EI}\frac{^{4}}{x^{4}}}} & (717) \end{matrix}$

Then, we obtain the following differential equation:

Lu(x)=f(x)  (718)

When boundary conditions are given as

[Formula 327]

u(0)=u _(o)

u&(0)=θ_(o)

−EI&&(0)=M _(o)

−EI&&&(0)=F _(o)  (719)

we obtain the following analytical solution:

$\begin{matrix} {{u(x)} = {{\frac{1}{EI}{\int_{0}^{x}{\int_{0}^{\xi}{\int_{0}^{\xi}{\int_{0}^{\xi}{{f(\xi)}\ {\xi}\ {\xi}\ {\xi}\ {\xi}}}}}}} - {\frac{1}{6}\frac{1}{EI}F_{o}x^{3}} - {\frac{1}{2}\frac{1}{EI}M_{o}x^{2}} + {\theta_{o}x} + u_{o}}} & (720) \end{matrix}$

When boundary conditions are given as:

u(0)=u _(o)

u&(0)=θ_(o)

u(λ)=u _(λ)

u&(λ)=θ_(λ)  (721)

we obtain the following analytical solution:

$\begin{matrix} {{u(x)} = {{\frac{1}{EI}\begin{pmatrix} {{\int_{0}^{x}{\int_{0}^{\xi}{\int_{0}^{\xi}{\int_{0}^{\xi}{{f(\xi)}\ {\xi}\ {\xi}\ {\xi}\ {\xi}}}}}} +} \\ {\left( \frac{x}{\lambda} \right)^{2}\begin{pmatrix} {{\left( {{2\frac{x}{\lambda}} - 3} \right){\int_{0}^{\lambda}{\int_{0}^{\xi}{\int_{0}^{\xi}{\int_{0}^{\xi}{{f(\xi)}\ {\xi}\ {\xi}\ {\xi}\ {\xi}}}}}}} -} \\ {\left( {x - \lambda} \right){\int_{0}^{\lambda}{\int_{0}^{\xi}{\int_{0}^{\xi}{{f(\xi)}{\xi}\ {\xi}\ {\xi}}}}}} \end{pmatrix}} \end{pmatrix}} + {\left( {\frac{x}{\lambda} - 1} \right)^{2}\left( {{2\frac{x}{\lambda}} + 1} \right)u_{o}} + {\left( {\frac{x}{\lambda} - 1} \right)^{2}x\; \theta_{o}} - {\left( \frac{x}{\lambda} \right)^{2}\left( {{2\frac{x}{\lambda}} - 3} \right)u_{\lambda}} + {\left( \frac{x}{\lambda} \right)^{2}\left( {x - \lambda} \right)\theta_{\lambda}}}} & (722) \end{matrix}$

[Formula 328]

Regarding a distributed load f(x), let a constant having dimensionality of (force/length) be Q, and let a dimensionless function be p(x). Then, the distributed load is given as:

f(x)≡Q·p(x)  (723)

Since terms having dimensionality of force are

$\frac{EI}{l^{2}}$

and Q_(l),

$\begin{matrix} \frac{Q\; \lambda^{3}}{EI} & (724) \end{matrix}$

is dimensionless. A dimensionless position ξ is given as:

$\begin{matrix} {\xi \equiv \frac{x}{\lambda}} & (725) \end{matrix}$

Dimensionless displacements are defined as:

$\begin{matrix} {{{\hat{u}(x)} \equiv {\frac{EI}{Q\; \lambda^{4}}{u(x)}}}{{{\hat{u}}_{o} \equiv {\frac{EI}{Q\; \lambda^{4}}u_{o}}},{{\hat{u}}_{\lambda} \equiv {\frac{EI}{Q\; \lambda^{4}}u_{\lambda}}}}} & (726) \end{matrix}$

Dimensionless angles are defined as:

$\begin{matrix} {{{\hat{\theta}}_{o} \equiv {\frac{EI}{Q\; \lambda^{3}}\theta_{o}}},{{\hat{\theta}}_{\lambda} \equiv {\frac{EI}{Q\; \lambda^{3}}\theta_{\lambda}}}} & (727) \end{matrix}$

Dimensionless moments are defined as:

$\begin{matrix} {{{\hat{M}}_{o} \equiv {\frac{1}{Q\; \lambda^{2}}M_{o}}},{{\hat{M}}_{\lambda} \equiv {\frac{1}{Q\; \lambda^{2}}M_{\lambda}}}} & (728) \end{matrix}$

Dimensionless shear forces are defined as:

$\begin{matrix} {{{\hat{F}}_{o} \equiv {\frac{1}{Q\; \lambda}F_{o}}},{{\hat{F}}_{\lambda} \equiv {\frac{1}{Q\; \lambda}F_{\lambda}}}} & (729) \end{matrix}$

10.4.2 Adjoint Boundary Condition and Adjoint Differential Operators [Formula 329]

Multiplying eq. (718) by u* and integrating the same, we obtain:

∫₀ ^(λ) Lu·u*dx=∫ ₀ ^(λ) f·u*dx  (730)

By partial integration of the left side of the equation, we obtain:

$\begin{matrix} {\mspace{20mu} \left\lbrack {{Formula}\mspace{14mu} 330} \right\rbrack} & \; \\ {{{\int_{0}^{\lambda}{{{Lu} \cdot u^{*}}{x}}} = {{{EI}\left\lbrack {\text{?}^{*} - \text{?} + {u\text{?}} - {u\text{?}}} \right\rbrack}_{0}^{\lambda} + {\int_{0}^{\lambda}{{u \cdot \left( {{EI}\; \frac{^{4}}{x^{4}}} \right)}u^{*}{x}}}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (731) \end{matrix}$

Let the operator L* be:

$\begin{matrix} {L^{*} \equiv {{EI}\frac{^{4}}{x^{4}}}} & (732) \end{matrix}$

Let the boundary term R be:

$\begin{matrix} {\mspace{79mu} {{R \equiv {{EI}\left\lbrack {\text{?} - \text{?} + {u\text{?}} - {u\text{?}}} \right\rbrack}_{0}^{\lambda}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (733) \end{matrix}$

Then, eq. (731) is transformed to:

∫₀ ^(λ) Lu·u*dx=R+∫ ₀ ^(λ) u·L*u*dx  (734)

The operator satisfies:

L*=L  (735)

Therefore, this is a self-adjoint operator. It should be noted that the boundary term R is transformed with use of the following equations:

$\begin{matrix} {\mspace{79mu} {\begin{matrix} {\theta \equiv {u\text{?}}} & {\theta^{*} \equiv {u\mspace{11mu} \text{?}}} \\ {M \equiv {{- {EI}}\text{?}}} & {M^{*} \equiv {{- {EI}}\text{?}}} \\ {F \equiv {{- {EI}}\text{?}}} & {F^{*} \equiv {{- {EI}}\text{?}}} \end{matrix}{\text{?}\text{indicates text missing or illegible when filed}}}} & (736) \end{matrix}$

The transformed boundary term R is given as:

R≡[−Fu*+Mθ*−θM*+uF*] ₀ ^(λ)  (737)

[Formula 331] [Exemplary Non-Self-Adjoint Boundary Condition 1]

Regarding the boundary term R, the boundary conditions are given as:

u(0)=0

θ(0)=0

M(0)=0

F(0)=0  (738)

Then, we obtain the following as adjoint boundary conditions:

u*(λ)=0

θ*(λ)=0

M*(λ)=0

F*(λ)=0  (739)

As the conditions of eq. (738) and (739) are different, this is a non-self-adjoint boundary condition.

[Exemplary Self-Adjoint Boundary Condition 1]

Regarding the boundary term R, the boundary conditions are given as:

u(0)=0

θ(0)=0

u(λ)=0

θ(λ)=0  (740)

Then, we obtain the following as adjoint boundary conditions:

u*(0)=0

θ*(0)=0

u*(λ)=0

θ*(λ)=0  (741)

As the conditions of eq. (740) and eq. (741) coincide, this is a self-adjoint boundary condition.

10.4.3 Homogenization of Boundary Condition [Formula 332]

An index B is added to a term that satisfies an inhomogeneous boundary condition so as to let the term be u_(B), and an index H is added a term that satisfies a homogeneous boundary condition so as to let the term be u_(H). The displacement u is expressed as follows, with the sum of these:

u(x)≡u _(B)(x)+u _(H)(x)  (742)

[Exemplary Non-Self-Adjoint Boundary Condition 1]

In the case where the boundary conditions are

[Formula 333]

u(0)=u

θ(0)=θ_(o)

M(0)=M _(o)

F(0)=F _(o,)  (743)

we define the boundary function u_(B) as

$\begin{matrix} {{u_{B}(x)} \equiv {{\frac{1}{2\;}\begin{bmatrix} {{{\lambda\left( {\theta_{o} + {F_{o}\frac{\lambda^{2}}{EI}}} \right)}\sin \; \frac{x}{\lambda}} + {{\lambda\left( {\theta_{o} - {F_{o}\frac{\lambda^{2}}{EI}}} \right)}\sinh \; \frac{x}{\lambda}} +} \\ {{\left( {u_{o} + {M_{o}\; \frac{\lambda^{2}}{EI}}} \right)\cos \; \frac{x}{\lambda}} + {\left( {u_{o} - {M_{o}\; \frac{\lambda^{2}}{EI}}} \right)\cosh \; \frac{x}{\lambda}}} \end{bmatrix}}.}} & (744) \end{matrix}$

Accordingly, we obtain:

u _(B)(0)=u _(o)

u&_(B)(0)=θ_(o)

−EI

(0)=M _(o)

−EI

(0)=F _(o)  (745)

Therefore, the following homogeneous boundary conditions are imposed on

u _(H)(0)=0

u

_(H)(0)=0

u

_(H)(0)=0

u

_(H)(0)=0  (746)

This example becomes a non-self-adjoint problem.

It should be noted that the following is established, which is a useful property:

$\begin{matrix} {{Lu}_{B} = {{\left( {{EI}\; \frac{^{4}}{x^{4}}} \right)u_{B}} = {\frac{EI}{\lambda^{4}}u_{B}}}} & (747) \end{matrix}$

[Formula 334] [Exemplary Self-Adjoint Boundary Condition 1]

In the case where the boundary conditions are

u(0)=u _(o)

θ(0)=θ_(o)

u(λ)=u _(λ)

θ(λ)=θ_(λ)  (748)

we define the boundary function u_(B) as

$\begin{matrix} {{u_{B}(x)} \equiv {\frac{1}{2}\begin{bmatrix} {\frac{{\cos \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {u_{o} + u_{\lambda}} \right)\sinh \; \frac{1}{2}} + {{\lambda \left( {\theta_{o} - \theta_{\lambda}} \right)}\cosh \; \frac{1}{2}}} \right)}{{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} + {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}} +} \\ {\frac{{\cosh \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {u_{o} + u_{\lambda}} \right)\sin \; \frac{1}{2}} - {{\lambda \left( {\theta_{o} - \theta_{\lambda}} \right)}\cos \; \frac{1}{2}}} \right)}{\; {{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} + {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}}} -} \\ {\frac{{\sinh \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {u_{o} - u_{\lambda}} \right)\cos \; \frac{1}{2}} + {{\lambda \left( {\theta_{o} + \theta_{\lambda}} \right)}\sin \; \frac{1}{2}}} \right)}{{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} - {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}} +} \\ \frac{{\sin \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {u_{o} - u_{\lambda}} \right)\cosh \; \frac{1}{2}} + {{\lambda \left( {\theta_{o} + \theta_{\lambda}} \right)}\sinh \; \frac{1}{2}}} \right)}{{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} - {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}} \end{bmatrix}}} & (749) \end{matrix}$

Accordingly, we obtain:

u _(B)(0)=u _(o)

u&_(B)(0)=θ_(o)

u _(B)(λ)=u _(λ)

u&_(B)(λ)=θ_(λ)  (750)

Therefore, the following homogeneous boundary conditions are imposed on

u _(H)(0)=0

u&_(H)(0)=0

u _(H)(λ)=0

u&_(H)(λ)=0  (751)

This example becomes a self-adjoint problem.

It should be noted that the following is established, which is a useful property:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 335} \right\rbrack & \; \\ {{Lu}_{B} = {{\left( {{EI}\frac{^{4}}{x^{4}}} \right)u_{B}} = {\frac{EI}{\lambda^{4}}u_{B}}}} & (752) \end{matrix}$

When eq. (742) is substituted into eq. (718), the differential equation (718) is transformed to:

Lu _(H) =f _(H)  (753)

where

f _(H) ≡f−Lu _(B)  (754)

This is equivalent to eqs. (40) and (41), and the problem is changed to a problem of determining a solution function U_(H) according to the boundary function u_(B).

10.4.4 Eigenfunction Sets [Formula 336]

In the simultaneous eigenvalue problem of

$\begin{matrix} \left\{ {\begin{matrix} {{L\; \varphi} = {\lambda \; w\; \varphi^{*}}} \\ {{L^{*}\varphi^{*}} = {\lambda \; w\; \varphi}} \end{matrix},} \right. & (755) \end{matrix}$

the weight w is given as:

$\begin{matrix} {w \equiv \frac{EI}{\lambda^{4}\;}} & (756) \end{matrix}$

From the simultaneous eigenvalue problem, the following equations on each eigenfunction are obtained:

$\begin{matrix} {{L^{*}L\; \varphi} = {{\lambda^{2}\left( \frac{EI}{\lambda^{4}} \right)}^{2}\varphi}} & (757) \\ {{{LL}^{*}\varphi^{*}} = {{\lambda^{2}\left( \frac{EI}{\lambda^{4\;}} \right)}^{2}\varphi^{*}}} & (758) \end{matrix}$

The operators satisfy:

$\begin{matrix} {{L^{*}L} = {{LL}^{*} = {({EI})^{2}\frac{^{8}}{x^{8}}}}} & (759) \end{matrix}$

Since eqs. (757) and (758) take the same form, eq. (757) may be solved so that the formats of φ,φ* can be obtained. Eq. (757) is given as:

$\begin{matrix} {{\left( {({EI})^{2}\frac{^{8}}{x^{8}}} \right)\varphi} = {{\lambda^{2}\left( \frac{EI}{\lambda^{4}} \right)}^{2}\varphi}} & (760) \end{matrix}$

And it is transformed to:

$\begin{matrix} {{\left( {\frac{^{4}}{x^{4}} - {\frac{1}{\lambda^{4}}\lambda}} \right)\left( {\frac{^{4}}{x^{4}} + {\frac{1}{\lambda^{4}}\lambda}} \right)\varphi} = 0} & (761) \end{matrix}$

[Formula 337]

This solution has a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈, a₁*, a₂*, a₃*, a₄*, a₅*, a₆*, a₇*, a₈* as arbitrary coefficients. Given

$\begin{matrix} {{\omega_{1} \equiv {\frac{1}{\lambda}\sqrt[4]{\lambda}}}{{\omega_{2} \equiv {\frac{1}{\sqrt{2}}\omega_{1}}},}} & (762) \end{matrix}$

we obtain:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {{a_{1}\sin \; \omega_{1}x} + {a_{2}\cos \; \omega_{1}x} + {a_{3}\sinh \; \omega_{1}x} + {a_{4}\cosh \; \omega_{1}x} +}} \\ {{a_{5}\sin \; \omega_{2}x\; \sinh \; \omega_{2}x} + {a_{6}\cos \; \omega_{2}x\; \sinh \; \omega_{2}x} +} \\ {{a_{7}\sin \; \omega_{2}x\; \cosh \; \omega_{2}x} + {a_{8}\cos \; \omega_{2}x\; \cosh \; \omega_{2}x}} \\ {\varphi^{*} = {{a_{1}^{*}\sin \; \omega_{1}x} + {a_{2}^{*}\cos \; \omega_{1}x} + {a_{3}^{*}\sinh \; \omega_{1}x} + {a_{4}^{*}\cosh \; \omega_{1}x} +}} \\ {{a_{5}^{*}\sin \; \omega_{2}x\; \sinh \; \omega_{2}x} + {a_{6}^{*}\cos \; \omega_{2}x\; \sinh \; \omega_{2}x} +} \\ {{a_{7}^{*}\sin \; \omega_{2}x\; \cosh \; \omega_{2}x} + {a_{8}^{*}\cos \; \omega_{2}x\; \cosh \; \omega_{2}x}} \end{matrix} \right. & (763) \end{matrix}$

Transforming the simultaneous eigenvalue problem (755), we obtain:

$\begin{matrix} \left\{ \begin{matrix} {{\left( \frac{^{4}}{x^{4}} \right)\varphi} = {\lambda \frac{1}{\lambda^{4}}\varphi^{*}}} \\ {{\left( \frac{^{4}}{x^{4}} \right)\varphi^{*}} = {\lambda \frac{1}{\lambda^{4}}\varphi}} \end{matrix} \right. & (764) \end{matrix}$

The coefficients a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈, a₁*, a₂*, a₃*, a₄*, a₅*, a₆*, a₇*, a₈* can be decided so as to satisfy eq. (764). As a result of decision of the coefficients, the combinations we obtain are the following eight combinations:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {\cos \; \omega_{1}x}} \\ {\varphi^{*} = {\cos \; \omega_{1}x}} \end{matrix} \right. & (765) \\ \left\lbrack {{Formula}\mspace{14mu} 338} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi = {\sin \; \omega_{1}x}} \\ {\varphi^{*} = {\sin \; \omega_{1}x}} \end{matrix} \right. & (766) \\ \left\{ \begin{matrix} {\varphi = {\cosh \; \omega_{1}x}} \\ {\varphi^{*} = {\cosh \; \omega_{1}x}} \end{matrix} \right. & (767) \\ \left\{ \begin{matrix} {\varphi = {\sinh \; \omega_{1}x}} \\ {\varphi^{*} = {\sinh \; \omega_{1}x}} \end{matrix} \right. & (768) \\ \left\{ \begin{matrix} {\varphi = {\cos \; \omega_{2}x\; \cosh \; \omega_{2}x}} \\ {\varphi^{*} = {{- \cos}\; \omega_{2}x\; \cosh \; \omega_{2}x}} \end{matrix} \right. & (769) \\ \left\{ \begin{matrix} {\varphi = {\cos \; \omega_{2}x\; \sinh \; \omega_{2}x}} \\ {\varphi^{*} = {{- \cos}\; \omega_{2}x\; \sinh \; \omega_{2}x}} \end{matrix} \right. & (770) \\ \left\{ \begin{matrix} {\varphi = {\sin \; \omega_{2}x\; \cosh \; \omega_{2}x}} \\ {\varphi^{*} = {{- \sin}\; \omega_{2}x\; \cosh \; \omega_{2}x}} \end{matrix} \right. & (771) \\ \left\{ \begin{matrix} {\varphi = {\sin \; \omega_{2}x\; \sinh \; \omega_{2}x}} \\ {\varphi^{*} = {{- \sin}\; \omega_{2}x\; \sinh \; \omega_{2}x}} \end{matrix} \right. & (772) \end{matrix}$

These combinations only satisfy the simultaneous differential equations (764), and such conditions that these combinations should satisfy the boundary condition have not been imposed.

10.4.5 Eigenfunction [Formula 339] [Exemplary Non-Self-Adjoint Boundary Condition 1]

In the case where the homogeneous boundary condition is expressed by eq. (746), the following conditions are imposed:

$\begin{matrix} {\mspace{79mu} \left\{ {\begin{matrix} {{\varphi (0)} = 0} \\ {{\text{?}(0)} = 0} \\ {{\text{?}(0)} = 0} \\ {{\text{?}(0)} = 0} \\ {{\varphi^{*}(\lambda)} = 0} \\ {{\text{?}(\lambda)} = 0} \\ {{\text{?}(\lambda)} = 0} \\ {{\text{?}(\lambda)} = 0} \end{matrix}\text{?}\text{indicates text missing or illegible when filed}} \right.} & (773) \end{matrix}$

The characteristic equation and the eigenfunctions obtained consequently are relatively long, and cannot be described herein. Studies are going on so as to simplify the same. Numerical values of the same can be obtained. The first four eigenvalues are:

$\begin{matrix} {\lambda = \left\{ \begin{matrix} 45.018 \\ 630.33 \\ 3889.2 \\ 14614. \end{matrix} \right.} & (774) \end{matrix}$

Each eigenfunction is normalized so that the maximum value is 1, and is shown in FIG. 33.

[Exemplary Self-Adjoint Boundary Condition 1] In the case where the homogeneous boundary condition is expressed by eq. (751), the following conditions are imposed:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 340} \right\rbrack} & \; \\ {\mspace{79mu} \left\{ \begin{matrix} {{\varphi (0)} = 0} \\ {{\text{?}(0)} = 0} \\ {{\varphi (\lambda)} = 0} \\ {{\text{?}(\lambda)} = 0} \\ {{\varphi^{*}(0)} = 0} \\ {{\text{?}(0)} = 0} \\ {{\varphi^{*}(\lambda)} = 0} \\ {{\text{?}(\lambda)} = 0} \end{matrix} \right.} & (775) \\ {\text{?}\text{indicates text missing or illegible when filed}} & \; \end{matrix}$

As a result, we obtain the following characteristic equation:

−1+cos ω₁λ cos hω ₁λ=0  (776)

With respect to λ that satisfies this characteristic equation, the following function is an eigenfunction:

$\begin{matrix} \left\{ \begin{matrix} {\varphi = {\left( {{\sinh \; \omega_{1}\lambda} - {\sin \; \omega_{1}\lambda}} \right)\left( {{\cosh \; \omega_{1}x} - {\cos \; \omega_{1}x}} \right)}} \\ {{- \left( {{\cosh \; \omega_{1}\lambda} - {\cos \; \omega_{1}\lambda}} \right)}\left( {{\sinh \; \omega_{1}x} - {\sin \; \omega_{1}x}} \right)} \end{matrix} \right. & (777) \end{matrix}$

When eqs. (725) and (762) are substituted into these, the characteristic equation is transformed to a function of λ, and the eigenfunction is transformed to a function of λ,ξ. Then, we obtain the first four eigenvalues as follows:

$\begin{matrix} {\lambda = \left\{ \begin{matrix} 500.56 \\ 3803.5 \\ 14618. \\ 39944. \end{matrix} \right.} & (778) \end{matrix}$

Each eigenfunction is normalized so that the maximum value is 1, and is shown in FIG. 34.

10.4.6 Eigenfunction Method [Formula 341] [Exemplary Non-Self-Adjoint Boundary Condition 1]

The solution displacement u(x) is given as:

$\begin{matrix} {\mspace{79mu} {{{u(x)} \equiv {{u_{B}(x)} + {u_{H}(x)}}}{{u_{B}(x)} \equiv {\frac{1}{2}\begin{bmatrix} {{{\lambda \left( {\theta_{o} + {F_{o}\frac{\lambda^{2}}{EI}}} \right)}\sin \frac{x}{\lambda}} + {{\lambda \left( {\theta_{o} - {F_{o}\frac{\lambda^{2}}{EI}}} \right)}\sinh \frac{x}{\lambda}} +} \\ {{\left( {u_{o} + {M_{o}\frac{\lambda^{2}}{EI}}} \right)\cos \frac{x}{\lambda}} + {\left( {u_{o} - {M_{o}\frac{\lambda^{2}}{EI}}} \right)\cosh \frac{x}{\lambda}}} \end{bmatrix}}}\mspace{20mu} {{u_{H}(x)} \equiv {\sum\limits_{i}{c_{i}{\varphi_{i}(x)}}}}\mspace{20mu} {c_{i} \equiv {\frac{1}{\lambda_{i}}\left( {{\frac{Q\; \lambda^{4}}{EI}\frac{\int_{0}^{\lambda}{{p \cdot \varphi_{i}^{*}}\ {x}}}{\int_{0}^{\lambda}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {x}}}} - \frac{\int_{0}^{\lambda}{{u_{B} \cdot \varphi_{i}^{*}}\ {x}}}{\int_{0}^{\lambda}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {x}}}} \right)}}}} & (779) \end{matrix}$

Making this dimensionless gives:

$\begin{matrix} {\mspace{79mu} {{{{{\hat{u}(x)} \equiv {{{\hat{u}}_{B}(x)} + {{\hat{u}}_{H}(x)}}}{{{\hat{u}}_{B}(x)} \equiv {\frac{1}{2}\begin{bmatrix} {{\left( {{\hat{\theta}}_{o} + {\hat{F}}_{o}} \right)\sin \frac{x}{\lambda}} + {\left( {{\hat{\theta}}_{o} - {\hat{F}}_{o}} \right)\sinh \frac{x}{\lambda}} +} \\ {{\left( {{\hat{u}}_{o} + {\hat{M}}_{o}} \right)\cos \frac{x}{\lambda}} + {\left( {{\hat{u}}_{o} - {\hat{M}}_{o}} \right)\cosh \frac{x}{\lambda}}} \end{bmatrix}}}\mspace{20mu} {{\hat{u}}_{H}(x)}} \equiv {\sum\limits_{i}{{\hat{c}}_{i}{\varphi_{i}(x)}}}}\mspace{20mu} {{\hat{c}}_{i} \equiv {\frac{1}{\lambda_{i}}\left( {\frac{\int_{0}^{\lambda}{{p \cdot \varphi_{i}^{*}}\ {x}}}{\int_{0}^{\lambda}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {x}}} - \frac{\int_{0}^{\lambda}{{{\hat{u}}_{B} \cdot \varphi_{i}^{*}}\ {x}}}{\int_{0}^{\lambda}{{\varphi_{i}^{*} \cdot \varphi_{i}^{*}}\ {x}}}} \right)}}}} & (780) \end{matrix}$

[Formula 342] [Exemplary Self-Adjoint Boundary Condition 1]

A solution displacement u(x) is given as:

$\begin{matrix} {\mspace{79mu} {{{u(x)} \equiv {{u_{B}(x)} + {u_{H}(x)}}}{{u_{B}(x)} \equiv {\frac{1}{2}\begin{bmatrix} {\frac{{\cos \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {u_{o} + u_{\lambda}} \right)\sinh \frac{1}{2}} + {{\lambda \left( {\theta_{o} - \theta_{\lambda}} \right)}\cosh \frac{1}{2}}} \right)}{{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} + {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}} +} \\ {\frac{{\cosh \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {u_{o} + u_{\lambda}} \right)\sin \frac{1}{2}} - {{\lambda \left( {\theta_{o} - \theta_{\lambda}} \right)}\cos \frac{1}{2}}} \right)}{{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} + {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}} -} \\ {\frac{{\sinh \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {u_{o} - u_{\lambda}} \right)\cos \frac{1}{2}} + {{\lambda \left( {\theta_{o} + \theta_{\lambda}} \right)}\sin \frac{1}{2}}} \right)}{{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} - {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}} +} \\ \frac{{\sin \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {u_{o} - u_{\lambda}} \right)\cosh \frac{1}{2}} + {{\lambda \left( {\theta_{o} + \theta_{\lambda}} \right)}\sinh \frac{1}{2}}} \right)}{{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} - {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}} \end{bmatrix}}}\mspace{20mu} {{u_{H}(x)} \equiv {\sum\limits_{i}{c_{i}{\varphi_{i}(x)}}}}\mspace{20mu} {c_{i} \equiv {\frac{1}{\lambda_{i}}\left( {{\frac{Q\; \lambda^{4}}{EI}\frac{\int_{0}^{\lambda}{{p \cdot \varphi_{i}^{*}}\ {x}}}{\int_{0}^{\lambda}{{\varphi_{i} \cdot \varphi_{i}}\ {x}}}} - \frac{\int_{0}^{\lambda}{{u_{B} \cdot \varphi_{i}}\ {x}}}{\int_{0}^{\lambda}{{\varphi_{i} \cdot \varphi_{i}}\ {x}}}} \right)}}}} & (781) \end{matrix}$

Making this dimensionless gives:

$\begin{matrix} {\mspace{79mu} {{{\hat{u}(x)} \equiv {{{\hat{u}}_{B}(x)} + {{\hat{u}}_{H}(x)}}}{{u_{B}(x)} \equiv {\frac{1}{2}\begin{bmatrix} {\frac{{\cos \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {{\hat{u}}_{o} + u_{\lambda}} \right)\sinh \frac{1}{2}} + {\left( {{\hat{\theta}}_{o} - {\hat{\theta}}_{\lambda}} \right)\cosh \frac{1}{2}}} \right)}{{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} + {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}} +} \\ {\frac{{\cosh \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {{\hat{u}}_{o} + {\hat{u}}_{\lambda}} \right)\sin \frac{1}{2}} - {\left( {{\hat{\theta}}_{o} - {\hat{\theta}}_{\lambda}} \right)\cos \frac{1}{2}}} \right)}{{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} + {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}} -} \\ {\frac{{\sinh \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {{\hat{u}}_{o} - {\hat{u}}_{\lambda}} \right)\cos \frac{1}{2}} + {\left( {{\hat{\theta}}_{o} + {\hat{\theta}}_{\lambda}} \right)\sin \frac{1}{2}}} \right)}{{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} - {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}} +} \\ \frac{{\sin \left( {\frac{1}{2} - \frac{x}{\lambda}} \right)}\left( {{\left( {{\hat{u}}_{o} - {\hat{u}}_{\lambda}} \right)\cosh \frac{1}{2}} + {\left( {{\hat{\theta}}_{o} + {\hat{\theta}}_{\lambda}} \right)\sinh \frac{1}{2}}} \right)}{{\cosh \; \frac{1}{2}\sin \; \frac{1}{2}} - {\cos \; \frac{1}{2}\sinh \; \frac{1}{2}}} \end{bmatrix}}}\mspace{20mu} {{{\hat{u}}_{H}(x)} \equiv {\sum\limits_{i}{{\hat{c}}_{i}{\varphi_{i}(x)}}}}\mspace{20mu} {{\hat{c}}_{i} \equiv {\frac{1}{\lambda_{i}}\left( {\frac{\int_{0}^{\lambda}{{p \cdot \varphi_{i}}\ {x}}}{\int_{0}^{\lambda}{{\varphi_{i} \cdot \varphi_{i}}\ {x}}} - \frac{\int_{0}^{\lambda}{{{\hat{u}}_{B} \cdot \varphi_{i}}\ {x}}}{\int_{0}^{\lambda}{{\varphi_{i} \cdot \varphi_{i}}\ {x}}}} \right)}}}} & (782) \end{matrix}$

10.4.7 Exemplary Non-Self-Adjoint Analysis 1 [Formula 343]

The dimensionless function p(x) of eq. (723) is given as:

$\begin{matrix} {{p(x)} \equiv {\sin \frac{\pi}{\lambda}x}} & (783) \end{matrix}$

According to eq. (720), we obtain an analytical solution as follows:

$\begin{matrix} {{\hat{u}(\xi)} = {\frac{{{- 6}{\pi\xi}} + {\pi^{3}\xi^{3}} + {6\; {\sin \left( {\pi \; \xi} \right)}}}{6\pi^{4}} - {\frac{\xi^{3}}{6}{\hat{F}}_{o}} - {\frac{\xi^{2}}{2}{\hat{M}}_{o}} + {\hat{u}}_{o} + {{\hat{\theta}}_{o}\xi}}} & (784) \end{matrix}$

Let the boundary condition be:

û _(o)=0, {circumflex over (θ)}_(o)=−0.01, {circumflex over (M)} _(o)=0, {circumflex over (F)} _(o)=0  (785)

Comparison between the analytical solutions of eq. (784) and the results obtained by the eigenfunction method of eq. (780) is shown in FIG. 35. In order to direct the curves in the same direction as that in FIG. 32, the drawings are created by inverting the sign of each function value. In reading the drawing, signs of values of only the vertical axis coordinates should be inverted. In the drawing, the broken lines indicate the analytical solutions, and the black solid lines indicate the results obtained by the eigenfunction method. “32M” means that the calculation is performed by using up to 32 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement û obtained by the calculation using about 3 modes by the eigenfunction method coincides well with the analytical solution.

10.4.8 Exemplary Non-Self-Adjoint Analysis 2 [Formula 344]

The dimensionless function p(x) of eq. (723) is given as:

$\begin{matrix} {{p(x)} \equiv {\cos \frac{\pi}{\lambda}x}} & (786) \end{matrix}$

According to eq. (720), we obtain an analytical solution as follows:

$\begin{matrix} {{\hat{u}(\xi)} = {\frac{{- 2} + {\pi^{2}\xi^{2}} + {2{\cos \left( {\pi \; \xi} \right)}}}{2\pi^{4}} - {\frac{\xi^{3}}{6}{\hat{F}}_{o}} - {\frac{\xi^{2}}{2}{\hat{M}}_{o}} + {\hat{u}}_{o} + {{\hat{\theta}}_{o}\xi}}} & (787) \end{matrix}$

Let the boundary condition be:

û _(o)=0, {circumflex over (θ)}_(o)=−0.01, {circumflex over (M)} _(o)=0, {circumflex over (F)} _(o)=0  (788)

Comparison between the analytical solutions of eq. (787) and the results obtained by the eigenfunction method of eq. (780) is shown in FIG. 36. In order to direct the curves in the same direction as that in FIG. 32, the drawings are created by inverting the sign of each function value. In reading the drawing, signs of values of only the vertical axis coordinates should be inverted. In the drawing, the broken lines indicate the analytical solutions, and the black solid lines indicate the results obtained by the eigenfunction method. “32M” means that the calculation is performed by using up to 32 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement ti obtained by the calculation using about 3 modes by the eigenfunction method coincides well with the analytical solution.

10.4.9 Exemplary Self-Adjoint Analysis 1 [Formula 345]

The dimensionless function p(x) of eq. (723) is given as:

$\begin{matrix} {{p(x)} \equiv {\sin \frac{\pi}{\lambda}x}} & (789) \end{matrix}$

According to eq. (722), we obtain an analytical solution as follows:

$\begin{matrix} {{\hat{u}(\xi)} = {\frac{{{\pi \left( {\xi - 1} \right)}\xi} + {\sin ({\pi\xi})}}{\pi^{4}} + {\left( {\xi - 1} \right)^{2}\left( {1 + {2\xi}} \right){\hat{u}}_{o}} - {{\xi^{2}\left( {{2\xi} - 3} \right)}{\hat{u}}_{\lambda}} + {{\xi \left( {\xi - 1} \right)}^{2}{\hat{\theta}}_{o}} + {{\xi^{2}\left( {\xi - 1} \right)}{\hat{\theta}}_{\lambda}}}} & (790) \end{matrix}$

Let the boundary condition be:

û _(o)=0,{circumflex over (θ)}_(o)=−0.01,û_(λ)=0,{circumflex over (θ)}_(λ)=0  (791)

Comparison between the analytical solutions of eq. (790) and the results obtained by the eigenfunction method of eq. (782) is shown in FIG. 37. In order to direct the curves in the same direction as that in FIG. 32, the drawings are created by inverting the sign of each function value. In reading the drawing, signs of values of only the vertical axis coordinates should be inverted. In the drawing, the broken lines indicate analytical solutions and the solid lines indicate the results obtained by the eigenfunction method. “32M” means that the calculation is performed by using up to 32 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement û obtained by the calculation using about 3 modes by the eigenfunction method coincides well with the analytical solution.

10.4.10 Exemplary Self-Adjoint Analysis 2 [Formula 346]

The dimensionless function p(x) of eq. (723) is given as:

$\begin{matrix} {{p(x)} \equiv {\cos \frac{\pi}{\lambda}x}} & (792) \end{matrix}$

According to eq. (722), we obtain:

$\begin{matrix} {{\hat{u}(\xi)} = {\frac{{- 1} + {6\xi^{2}} - {4\xi^{3}} + {\cos ({\pi\xi})}}{\pi^{4}} + {\left( {1 - {3\xi^{2}} + {2\xi^{3}}} \right){\hat{u}}_{o}} + {{\xi^{2}\left( {3 - {2\xi}} \right)}{\hat{u}}_{\lambda}} + {{\xi \left( {\xi - 1} \right)}^{2}{\hat{\theta}}_{o}} + {{\xi^{2}\left( {\xi - 1} \right)}{\hat{\theta}}_{\lambda}}}} & (793) \end{matrix}$

Let the boundary condition be:

û _(o)=0, {circumflex over (θ)}_(o)=−0.01, û _(λ)=0, {circumflex over (θ)}λ=0  (794)

Comparison between the analytical solutions of eq. (793) and the results obtained by the eigenfunction method of eq. (782) is shown in FIG. 38. In order to direct the curves in the same direction as that in FIG. 32, the drawings are created by inverting the sign of each function value. In reading the drawing, signs of values of only the vertical axis coordinates should be inverted. In the drawing, the broken lines indicate the analytical solutions, and the black solid lines indicate the results obtained by the eigenfunction method. “32M” means that the calculation is performed by using up to 32 modes. We can see that as the number of modes is greater, the accuracy of the eigenfunction method is improved, and the displacement û obtained by the calculation using about 3 modes by the eigenfunction method coincides well with the analytical solution.

[Other Exemplary Analysis Results]

FIG. 39 shows one solution other than the solution shown in FIG. 10 among a plurality of solutions obtained in the analysis shown in FIG. 10 in the case where such a plurality of solutions are generated when the eigenvalue is zero. With such a plurality of solutions presented to a designer, the designer can make studies about a method that is feasible in terms of design.

FIGS. 40A and 40B show a plurality of exemplary solutions of the displacement obtained by calculation for a finite element in the case where both of a displacement and a surface force of a top face of a square plate on which uniform gravity is acting, are zero. In examples shown in FIGS. 40A and 40B, 6×6 finite elements composes a square. Circles indicate nodes. 10 modes in total are shown. Arbitrary combinations of these become solutions. For example, by calculation shown in Section 9.2, a plurality of solutions can be determined. Further, a configuration may be such that a mode coefficient is made in synchronization with a slider and a deformation diagram is displayed in response to adjustment of the mode coefficient made by a designer using the slider. This allows the designer to study what supporting method is available, and allows the designer to have more images. This makes it possible for the designer to narrow down the feasible supporting method, and thereafter perform calculation again.

[Others]

The calculation unit 1, the setting unit 11, the adjoint boundary condition calculation unit 12, the non-self-adjoint calculation unit 13, the self-adjoint determination unit 15, and the self-adjoint calculation unit 14 in Embodiments 1 to 6 described above are configured by a processor of a computer that reads a program and data from a memory and executes data processing according to the program. The calculation unit 1 may obtain design data, and boundary condition data by receiving input by a user via an input device and a user interface, or alternatively, may obtain the same by reading it from a memory. Differential equation data are preferably recorded preliminarily in a memory to which the calculation unit 1 can make access.

The function of the calculation unit 1 can be realized by a CPU of one computer, or can be realized by a plurality of computers that can perform data communication with one another. Further, a program that causes a computer to execute the processing of the calculation unit 1, and a non-transitory recording medium that stores the program are also included in embodiments of the present invention.

It should be noted that the present invention is not limited to Embodiments 1 to 6 described above. The present invention can be applied to any analysis using differential equations, for example, structure analysis, vibration analysis, electric field analysis, material analysis, fluid analysis, thermal analysis, acoustic field analysis, electromagnetic analysis, or circuit simulator.

11. Supplementary Description 11.1 Details of Expression Transformation in “Direct Variational Method and Least-Squares Method” in Section 5.2 11.1.1 Direct Variational Method of Primal Problem [Formula 347]

To the differential equation with homogeneous boundary condition (40)

$\begin{matrix} {{{\sum\limits_{j}\; {L_{ij}u_{Hj}}} = f_{Hi}},} & \underset{({Aforementioned})}{(40)} \end{matrix}$

the solution function u_(Hj) of eq. (69)

$\begin{matrix} {u_{Hj} \equiv {\sum\limits_{k}\; {c_{k}\varphi_{jk}}}} & \underset{({Aforementioned})}{(69)} \end{matrix}$

is substituted, and the primal simultaneous differential equation (105)

$\begin{matrix} {{\sum\limits_{j}\; {L_{ij}\varphi_{j}}} = {\lambda \; w_{i}\varphi_{i}^{*}}} & \underset{({Aforementioned})}{(105)} \end{matrix}$

is used. Then, we obtain:

$\begin{matrix} {{\sum\limits_{k}\; {c_{k}\lambda_{k}\; w_{i}\varphi_{ik}^{*}}} = f_{Hi}} & \underset{({Aforementioned})}{(109)} \end{matrix}$

Here, λ_(k) is a k-th eigenvalue. An inner product of both sides of this equation and the function φ_(i)* is determined, and orthogonality among normal orthogonality of the following eq. (104) is used:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 348} \right\rbrack & \; \\ {{\sum\limits_{i}\; {\int_{S}{w_{i}{\varphi_{im}^{*} \cdot \varphi_{in}^{*}}\ {s}}}} = {{\sum\limits_{i}\; {\int_{S}{{\varphi_{im} \cdot w_{i}}\varphi_{in}\ {s}}}} = \delta_{mn}}} & \underset{({Aforementioned})}{(104)} \end{matrix}$

Then, we obtain:

$\begin{matrix} {{c_{k}\lambda_{k}{\sum\limits_{i}\; {\int_{S}{w_{i}{\varphi_{ik}^{*} \cdot \varphi_{ik}^{*}}\ {s}}}}} = {\sum\limits_{i}\; {\int_{S}{{f_{Hi} \cdot \varphi_{ik}^{*}}\ {s}}}}} & \underset{({Aforementioned})}{(110)} \end{matrix}$

This causes the coefficient c_(k) to be settled. The normalization of eq. (104) gives:

$\begin{matrix} {c_{k} = {\frac{1}{\lambda_{k}}{\sum\limits_{i}\; {\int_{S}{{f_{Hi} \cdot \varphi_{ik}^{*}}\ {s}}}}}} & \underset{({Aforementioned})}{(111)} \end{matrix}$

We find that from an inner product of the external force term f_(Hi) and the dual eigenfunction φ_(i)*, the coefficient c_(k) is settled.

[Formula 349]

The combining of a plurality of the above-described procedures is equivalent to the determining of an inner product of the dual variation of the following eq. (121) and eq. (40):

$\begin{matrix} {{\delta \; u_{i}^{*}} = {{\delta \; u_{Hi}^{*}} \equiv {\sum\limits_{k}\; {\delta \; c_{k}^{*}\varphi_{ik}^{*}}}}} & \underset{({Aforementioned})}{(121)} \end{matrix}$

In specific calculation, the inner product of the dual variation δu_(i)* and eq. (40) is determined as follows:

$\begin{matrix} {{\sum\limits_{i}\; {\int_{S}{\sum\limits_{j}\; {L_{ij}{u_{Hj} \cdot \delta}\; u_{i}^{*}\ {s}}}}} = {\sum\limits_{i}\; {\int_{S}{{f_{Hi} \cdot \delta}\; {u\ }_{i}^{*}{s}}}}} & (795) \end{matrix}$

This is equivalent to an inner product of the dual variation δu_(i)* and eq. (109), which is as follows:

$\begin{matrix} {{\sum\limits_{k}\; {c_{k}\lambda_{k}{\sum\limits_{i}\; {\int_{S}{w_{i}{\varphi_{ik}^{*} \cdot \delta}\; u_{i}^{*}\ {s}}}}}} = {\sum\limits_{i}\; {\int_{S}{{f_{Hi} \cdot \delta}\; u_{i}^{*}\ {s}}}}} & (796) \end{matrix}$

Substituting eq. (121) into this equation and utilizing the orthogonality of eq. (104) gives:

$\begin{matrix} {{\sum\limits_{k}\; {c_{k}\lambda_{k}\delta \; c_{k}^{*}{\sum\limits_{i}\; {\int_{S}{w_{i}{\varphi_{ik}^{*} \cdot \varphi_{ik}^{*}}\ {s}}}}}} = {\sum\limits_{k}\; {\delta \; c_{k}^{*}{\sum\limits_{i}\; {\int_{S}{{f_{Hi} \cdot \varphi_{ik}^{*}}\ {s}}}}}}} & (797) \end{matrix}$

By normalization of eq. (104), this equation is transformed to:

$\begin{matrix} {{\sum\limits_{k}{c_{k}\lambda_{k}\delta \; c_{k}^{*}}} = {\sum\limits_{k}{\delta \; c_{k}^{*}{\sum\limits_{i}{\int_{S}^{\;}{{f_{Hi} \cdot \varphi_{ik}^{*}}\ {s}}}}}}} & (798) \end{matrix}$

This is nothing but a combination of a plurality of the equations (111). By creating eq. (796) with respect to various δu_(i)*, that is, various δc_(k)*, and solving the simultaneous equations algebraically, the coefficients c_(k) can be determined.

From these, we can find that an equation obtained by transforming eq. (795), which is

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 350} \right\rbrack & \; \\ {{{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{Hj}}} - f_{Hi}} \right) \cdot \delta}\; u_{i}^{*}\ {s}}}} = 0},} & (799) \end{matrix}$

indicates the direct variational method, and is equivalent to the eigenfunction method of eq. (111). Substituting the external force term f_(H) of eq. (41), which is

$\begin{matrix} {{f_{Hi} \equiv {f_{i} - {\sum\limits_{j}{L_{ij}u_{Bj}}}}},} & \begin{matrix} (41) \\ ({Aforementioned}) \end{matrix} \end{matrix}$

into eq. (799) gives:

$\begin{matrix} {{\sum\limits_{i}{\int_{S}{{\left\{ {{\sum\limits_{j}{L_{ij}\left( {u_{Bj} + u_{Hj}} \right)}} - f_{i}} \right\} \cdot \delta}\; u_{i}^{*}\ {s}}}} = 0.} & (800) \end{matrix}$

The displacement of this equation is expressed with the primal displacement u_(j) of the following eq. (24):

[Formula 351]

u _(j) ≡u _(Bj) +u _(Hj)  (24) (Aforementioned)

Then, we obtain:

$\begin{matrix} {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}^{*}\ {s}}}} = 0} & \begin{matrix} (122) \\ ({Aforementioned}) \end{matrix} \end{matrix}$

This is the direct variational method of the primal problem.

In the case of the self-adjoint problem, the primal and dual eigenfunctions φ_(i) and φ_(i)* coincide via eq. (107).

Therefore, eqs. (120) and (121) are equal to each other, and eq. (122) is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 352} \right\rbrack & \; \\ {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}\ {s}}}} = 0} & \begin{matrix} (123) \\ ({Aforementioned}) \end{matrix} \end{matrix}$

11.1.2 Least-Squares Method of Primal Problem

According to the primal simultaneous differential equation (105)

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 353} \right\rbrack & \; \\ {{\sum\limits_{j}{L_{ij}\varphi_{j}}} = {\lambda \; w_{i}\varphi_{i}^{*}}} & \begin{matrix} (105) \\ ({Aforementioned}) \end{matrix} \end{matrix}$

and the primal variation δu_(i) of eq. (120)

$\begin{matrix} {{{\delta \; u_{i}} = {{\delta \; u_{Hi}} \equiv {\sum\limits_{k}{\delta \; c_{k}\varphi_{ik}}}}},} & \begin{matrix} (120) \\ ({Aforementioned}) \end{matrix} \end{matrix}$

we obtain a sum of differential coefficients of the primal variation δu_(j) as follows:

$\begin{matrix} \begin{matrix} {{\sum\limits_{j}{L_{ij}\delta \; u_{j}}} = {\sum\limits_{k}{\delta \; c_{k}{\sum\limits_{j}{L_{ij}\varphi_{jk}}}}}} \\ {= {\sum\limits_{k}{\delta \; c_{k}\ \lambda_{k}w_{i}\varphi_{ik}^{*}}}} \end{matrix} & \begin{matrix} (124) \\ ({Aforementioned}) \end{matrix} \end{matrix}$

In the primal and dual eigenfunctions using the same weight w with respect to the every component, we can define:

[Formula 354]

δc _(k) *≡δc _(k)λ_(k) w _(i) , w _(i) =w  (125) (Aforementioned)

Therefore, according to this equation and eq. (121), we obtain the dual variation at: as follows:

$\begin{matrix} {{{\delta \; u_{i}^{*}} = {{\sum\limits_{k}{\delta \; c_{k}^{*}\varphi_{ik}^{*}}} = {\sum\limits_{k}{\delta \; c_{k}\ \lambda_{k}w_{i}\varphi_{ik}^{*}}}}},{w_{i} = w}} & (801) \end{matrix}$

As the right side of the equation coincides with that of eq. (124), we can recognize:

$\begin{matrix} {{\delta \; u_{i}^{*}} = {{\sum\limits_{j}{L_{ij}\delta \; u_{j}}} = {\delta {\sum\limits_{j}{L_{ij}u_{j}}}}}} & \begin{matrix} (126) \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Substituting this equation into eq. (122) gives:

$\begin{matrix} {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; {\sum\limits_{j}{L_{ij}u_{j}\ {s}}}}}} = 0} & \begin{matrix} (127) \\ ({Aforementioned}) \end{matrix} \end{matrix}$

In the case where f is dealt with as a known external force, eq. (127) is equivalent to a case where the functional Π is given as

$\begin{matrix} {\Pi \equiv {\sum\limits_{i}{\int_{S}{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right)^{2}{s}}}}} & \begin{matrix} (128) \\ ({Aforementioned}) \end{matrix} \end{matrix}$

and its variation is zero. In other words, the least-squares method is applied as the variational principle.

11.1.3 Direct Variational Method of Dual Problem

The solution function u_(Hj)* of the following eq. (70) is substituted to the following differential equation (66) with homogeneous boundary condition:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 355} \right\rbrack & \; \\ {{\sum\limits_{j}{L_{ji}^{*}u_{Hj}^{*}}} = f_{Hi}^{*}} & \begin{matrix} (66) \\ ({Aforementioned}) \end{matrix} \\ {u_{Hj}^{*} = {\sum\limits_{k}{c_{k}^{*}\varphi_{jk}^{*}}}} & \begin{matrix} (70) \\ ({Aforementioned}) \end{matrix} \end{matrix}$

And the following dual simultaneous differential equation (106) is used:

$\begin{matrix} {{\sum\limits_{j}{L_{ji}^{*}\ \varphi_{j}^{*}}} = {\lambda \; w_{i}\varphi_{i}}} & \begin{matrix} (106) \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Then, we obtain:

$\begin{matrix} {{\sum\limits_{k}^{\;}\; {c_{k}^{*}\lambda_{k}w_{i}\varphi_{ik}}} = f_{Hi}^{*}} & \underset{({Aforementioned})}{(113)} \end{matrix}$

Here, λ_(k) is a k-th eigenvalue. An inner product of both sides of eq. (113) and the function φ_(i) is determined, and orthogonality among normal orthogonality of the following eq. (104) is used:

$\begin{matrix} {{\sum\limits_{i}^{\;}\; {\int_{S}^{\;}{w_{i}{\varphi_{im}^{*} \cdot \varphi_{in}^{*}}\ {s}}}} = {{\sum\limits_{i}^{\;}{\int_{S}^{\;}{{\varphi_{im} \cdot w_{i}}\varphi_{in}\ {s}}}} = \delta_{mn}}} & \underset{({Aforementioned})}{(104)} \end{matrix}$

Then, we obtain:

$\begin{matrix} {{c_{k}^{*}\lambda_{k}{\sum\limits_{i}^{\;}{\int_{S}^{\;}{w_{i}{\varphi_{ik} \cdot \varphi_{ik}}\ {s}}}}} = {\sum\limits_{i}^{\;}{\int_{S}^{\;}{{f_{Hi}^{*} \cdot \varphi_{ik}}{s}}}}} & \underset{({Aforementioned})}{(114)} \end{matrix}$

This causes the coefficient c_(k)*to be settled. The normalization of eq. (104) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 356} \right\rbrack & \; \\ {c_{k}^{*} = {\frac{1}{\lambda_{k}}{\sum\limits_{i}^{\;}{\int_{S}^{\;}{{f_{Hi}^{*} \cdot \varphi_{ik}}\ {s}}}}}} & \underset{({Aforementioned})}{(115)} \end{matrix}$

We find that from an inner product of the external force term f_(H)* and the primal eigenfunction φ_(i), the coefficient c_(k)* is settled.

The combining of a plurality of the above-described procedures is equivalent to the determining of an inner product of the primal variation δu_(i) of the following eq. (120) and eq. (66):

$\begin{matrix} {{\delta \; u_{i}} = {{\delta \; u_{Hi}} \equiv {\sum\limits_{k}^{\;}\; {\delta \; c_{k}\varphi_{ik}}}}} & \underset{({Aforementioned})}{(120)} \end{matrix}$

In specific calculation, the inner product of the primal variation δu_(i) and eq. (66) is determined as follows:

$\begin{matrix} {{\sum\limits_{i}^{\;}{\int_{S}^{\;}{\sum\limits_{j}^{\;}{L_{ji}^{*}{u_{Hj}^{*} \cdot \delta}\; u_{i}\ {s}}}}} = {\sum\limits_{i}^{\;}{\int_{S}^{\;}{{f_{Hi}^{*} \cdot \delta}\; u_{i}\ {s}}}}} & (802) \end{matrix}$

This is equivalent to an inner product of the primal variation δu_(i) and eq. (113), which is as follows:

$\begin{matrix} {{\sum\limits_{k}^{\;}{c_{k}^{*}\lambda_{k}{\sum\limits_{i}^{\;}{\int_{S}^{\;}{w_{i}{\varphi_{ik} \cdot \delta}\; u_{i}\ {s}}}}}} = {\sum\limits_{i}^{\;}{\int_{S}^{\;}{{f_{Hi}^{*} \cdot \delta}\; u_{i}\ {s}}}}} & (803) \end{matrix}$

Substituting eq. (120) into this equation and utilizing the orthogonality of eq. (104) gives:

$\begin{matrix} {{\sum\limits_{k}^{\;}\; {c_{k}^{*}\lambda_{k}\delta \; c_{k}{\sum\limits_{i}^{\;}\; {\int_{S}^{\;}{w_{i}{\varphi_{ik} \cdot \varphi_{ik}}\ {s}}}}}} = {\sum\limits_{k}^{\;}\; {\delta \; c_{k}{\sum\limits_{i}^{\;}{\int_{S}^{\;}{{f_{Hi}^{*} \cdot \varphi_{ik}}\ {s}}}}}}} & (804) \end{matrix}$

By normalization of eq. (104), this equation is transformed to:

$\begin{matrix} {{\sum\limits_{k}^{\;}{c_{k}^{*}\lambda_{k}\delta \; c_{k}}} = {\sum\limits_{k}^{\;}{\delta \; c_{k}{\sum\limits_{i}^{\;}{\int_{S}^{\;}{{f_{Hi}^{*} \cdot \varphi_{ik}}\ {s}}}}}}} & (805) \end{matrix}$

This is nothing but a combination of a plurality of the equations (115). By creating the foregoing equations with respect to various δu_(i), that is, various δc_(k), and solving the simultaneous equations algebraically, the coefficients c_(k)* can be determined.

From these, we can find that an equation obtained by transforming eq. (802), which is

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 357} \right\rbrack & \; \\ {{{\sum\limits_{i}^{\;}{\int_{S}^{\;}{{\left( {{\sum\limits_{j}^{\;}\ {L_{ji}^{*}u_{Hj}^{*}}} - f_{Hi}^{*}} \right) \cdot \delta}\; u_{i}{s}}}} = 0},} & (806) \end{matrix}$

indicates the direct variational method, and is equivalent to the eigenfunction method of eq. (115). Substituting the external force term f_(H)* of eq. (67), which is

$\begin{matrix} {f_{Hi}^{*} \equiv {f_{i}^{*} - {\sum\limits_{j}^{\;}{L_{ji}^{*}u_{Bj}^{*}}}}} & \underset{({Aforementioned})}{(67)} \end{matrix}$

into eq. (806) gives:

$\begin{matrix} {{\sum\limits_{i}^{\;}{\int_{S}^{\;}{{\left\{ {{\sum\limits_{j}^{\;}{L_{ji}^{*}\left( {u_{Bj}^{*} + u_{Hj}^{*}} \right)}} - f_{i}^{*}} \right\} \cdot \delta}\; u_{i}\ {s}}}} = 0} & (807) \end{matrix}$

The displacement of this equation is expressed with the dual displacement u_(j)* of the following eq. (42):

u _(j) *≡u _(Bj) *+u _(Hj)*  (42) (Aforementioned)

Then, we obtain:

$\begin{matrix} {{\sum\limits_{i}^{\;}{\int_{S}^{\;}{{\left( {{\sum\limits_{j}^{\;}{L_{ji}^{*}u_{j}^{*}}} - f_{i}^{*}} \right) \cdot \delta}\; u_{i}\ {s}}}} = 0} & (808) \end{matrix}$

This is the direct variational method of the dual problem.

In the case of the self-adjoint problem, the primal and dual eigenfunctions φ_(i) and φ_(i)* coincide via eq. (107). Therefore, eq. (120) and (121) are equal to each other, and eq. (808) is transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 358} \right\rbrack & \; \\ {{\sum\limits_{i}^{\;}{\int_{S}^{\;}{{\left( {{\sum\limits_{j}^{\;}\ {L_{ji}^{*}u_{j}^{*}}} - f_{i}^{*}} \right) \cdot \delta}\; u_{i}^{*}{s}}}} = 0} & (809) \end{matrix}$

11.1.4 Least-Squares Method of Dual Problem

According to the dual simultaneous differential equation (106)

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 359} \right\rbrack & \; \\ {{\sum\limits_{j}^{\;}{L_{ji}^{*}\varphi_{j}^{*}}} = {\lambda \; w_{i}\varphi_{i}}} & \underset{({Aforementioned})}{(106)} \end{matrix}$

and the dual variation δu_(i)* of eq. (121)

$\begin{matrix} {{{\delta \; u_{i}^{*}} = {{\delta \; u_{Hi}^{*}} \equiv {\sum\limits_{k}{\delta \; c_{k}^{*}\varphi_{ik}^{*}}}}},} & \begin{matrix} {\mspace{130mu} (121)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

we obtain a sum of differential coefficients of dual variation δu_(j)* as follows:

$\begin{matrix} {{\sum\limits_{j}{L_{ji}^{*}\delta \; u_{j}^{*}}} = {{\sum\limits_{k}{\delta \; c_{k}^{*}{\sum\limits_{j}{L_{ji}^{*}\varphi_{jk}^{*}}}}} = {\sum\limits_{k}{\delta \; c_{k}^{*}\lambda_{k}w_{i}\varphi_{ik}}}}} & (810) \end{matrix}$

In the primal and dual eigenfunctions using the same weight w with respect to the every component, we can define:

δc _(k) ≡δc _(k)*λ_(k) w _(i) , w _(i) =w  (811)

Therefore, according to this equation and eq. (120), we obtain the primal variation ou, as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 360} \right\rbrack & \; \\ {{{\delta \; u_{i}} = {{\sum\limits_{k}{\delta \; c_{k}\varphi_{ik}}} = {\sum\limits_{k}{\delta \; c_{k}^{*}\lambda_{k}w_{i}\varphi_{k}}}}},{w_{i} = w}} & (812) \end{matrix}$

As the right side of the equation coincides with that of eq. (810), we can recognize:

$\begin{matrix} {{\delta \; u_{i}} = {{\sum\limits_{j}{L_{ji}^{*}\delta \; u_{j}^{*}}} = {\delta {\sum\limits_{j}{L_{ji}^{*}u_{j}^{*}}}}}} & (813) \end{matrix}$

Substituting this equation into eq. (808) gives:

$\begin{matrix} {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ji}^{*}u_{j}^{*}}} - f_{i}^{*}} \right) \cdot \delta}{\sum\limits_{j}{L_{ji}^{*}u_{j}^{*}\ {s}}}}}} = 0} & (814) \end{matrix}$

In the case where f_(i)* is dealt with a known external force, this equation is equivalent to a case where the functional Π is given as

$\begin{matrix} {\prod^{*}{\equiv {\sum\limits_{i}{\int_{S}{\left( {{\sum\limits_{j}{L_{ji}^{*}u_{j}^{*}}} - f_{i}^{*}} \right)^{2}\ {s}}}}}} & (815) \end{matrix}$

and its variation is zero. In other words, the least-squares method is applied as the variational principle.

11.2 Details of Equation Transformation in “Energy Principle in Theory of Elasticity” of Chapter 6 11.2.1 Formula of Partial Integration

Orthogonal coordinates x, y are taken in a two-dimensional region, and let a boundary surface of an object be C, and let an internal region of the body be S. Let components of an outward unit normal vector of a boundary surface be n_(x) and n_(y). Partial integration of arbitrary functions f, f* as functions of x, y gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 361} \right\rbrack & \; \\ {{\int_{S}{\frac{\partial f}{\partial x}f^{*}\ {s}}} = {{\int_{C}{n_{x}{f \cdot f^{*}}\ {c}}} - {\int_{S}{f\frac{\partial f^{*}}{\partial x}\ {s}}}}} & (816) \\ {and} & \; \\ {{\int_{S}{\frac{\partial f}{\partial y}f^{*}\ {s}}} = {{\int_{C}{n_{y}{f \cdot f^{*}}\ {c}}} - {\int_{S}{f\frac{\partial f^{*}}{\partial y}\ {{s}.}}}}} & (817) \end{matrix}$

11.2.2 Direct Variational Method of Self-Adjoint Problem

The direct variational method of the self-adjoint problem is expressed by the following eq. (123):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 362} \right\rbrack & \; \\ {{\sum\limits_{i}{\int_{S}\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}}\  \right)}}{{\delta \; u_{i}{s}} = 0}} & \begin{matrix} {\mspace{130mu} (123)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

The following description is supplementary explanation about the procedure of transforming this to obtain the following eq. (132):

δU=∫ _(C)(p _(x) δu _(x) +p _(y) δu _(y))dc+∫ _(S)(b _(x) δu _(x) +b _(y) δu _(y))ds  (132) (Aforementioned)

The differential equations of the theory of elasticity are expressed by the following eqs. (1) and (2):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 363} \right\rbrack & \; \\ {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} u_{x}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}u_{y}}} = {{- \frac{1}{G}}b_{x}}} & \begin{matrix} {\mspace{155mu} (1)} \\ ({Aforementioned}) \end{matrix} \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}u_{x}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} u_{y}}} = {{- \frac{1}{G}}b_{y}}} & \begin{matrix} {\mspace{155mu} (2)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

The original differential operator L_(ij) expressed by the following eq. (15):

$\begin{matrix} {\begin{bmatrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{bmatrix} \equiv {\quad\begin{bmatrix} {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} & {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \\ {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} & {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \end{bmatrix}}} & \begin{matrix} {\mspace{146mu} (15)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Defining the right sides of the differential equations (1) and (2) as follows

$\begin{matrix} {\begin{Bmatrix} f_{1} \\ f_{2} \end{Bmatrix} \equiv {{- \frac{1}{G}}\begin{Bmatrix} b_{x} \\ b_{y} \end{Bmatrix}}} & (818) \end{matrix}$

and describing eqs. (1) and (2) generally gives the following eq. (23):

$\begin{matrix} {{\sum\limits_{j}{L_{ij}u_{j}}} = f_{i}} & \begin{matrix} {\mspace{146mu} (23)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

With use of these equations, the left side of eq. (123) is transformed to:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 364} \right\rbrack} & \; \\ {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}\ {s}}}} \equiv {{\int_{S}{\left\lbrack {{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} u_{x}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}u_{y}} + {\frac{1}{G}b_{x}}} \right\rbrack \delta \; u_{x}\ {s}}} + {\int_{S}{\left\lbrack {{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}u_{x}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} u_{y}} + {\frac{1}{G}b_{y}}} \right\rbrack \delta \; u_{y}\ {s}}}}} & (819) \end{matrix}$

The stress components are expressed by the following eqs. (10), (11), and (12):

$\begin{matrix} {\sigma_{x} \equiv {G\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{x}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial u_{y}}{\partial y}}} \right\}}} & \begin{matrix} {\mspace{146mu} (10)} \\ ({Aforementioned}) \end{matrix} \\ {\sigma_{y} \equiv {G\left\{ {{\left( {\mu - 1} \right)\frac{\partial u_{x}}{\partial x}} + {\left( {\mu + 1} \right)\frac{\partial u_{y}}{\partial y}}} \right\}}} & \begin{matrix} {\mspace{146mu} (11)} \\ ({Aforementioned}) \end{matrix} \\ {\tau_{xy} = {\tau_{yx} \equiv {G\left( {\frac{\partial u_{y}}{\partial x} + \frac{\partial u_{x}}{\partial y}} \right)}}} & \begin{matrix} {\mspace{146mu} (12)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Expressing eq. (819) by using these stress components gives:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 365} \right\rbrack} & \; \\ {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}\ {s}}}} \equiv {{\frac{1}{G}{\int_{S}{\left( {\frac{\partial\sigma_{x}}{\partial x} + \frac{\partial\tau_{yx}}{\partial y} + b_{x}} \right)\delta \; u_{x}\ {s}}}} + {\frac{1}{G}{\int_{S}{\left( {\frac{\partial\tau_{xy}}{\partial x} + \frac{\partial\sigma_{y}}{\partial y} + b_{y}} \right)\delta \; u_{y}\ {s}}}}}} & (820) \end{matrix}$

According to the formula of partial integration (816), we obtain:

$\begin{matrix} {{{\int_{S}^{\;}{\frac{\partial\sigma_{x}}{\partial x}\delta \; u_{x}\ {s}}} = {{\int_{\; C}^{\;}{n_{x}{\sigma_{x} \cdot \delta}\; u_{x}\ {c}}} - {\int_{S}^{\;}{\sigma_{x}\frac{{\partial\delta}\; u_{x}}{\partial x}\ {s}}}}}{and}} & (821) \\ {{\int_{S}^{\;}{\frac{\partial\tau_{xy}}{\partial x}\delta \; u_{y}\ {s}}} = {{\int_{\; C}^{\;}{n_{x}{\tau_{xy} \cdot \delta}\; u_{y}\ {c}}} - {\int_{S}^{\;}{\tau_{xy}\frac{{\partial\delta}\; u_{y}}{\partial x}\ {s}}}}} & (822) \end{matrix}$

According to the formula of partial integration (817), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 366} \right\rbrack & \; \\ {{{\int_{S}^{\;}{\frac{\partial\tau_{yx}}{\partial y}\delta \; u_{x}\ {s}}} = {{\int_{\; C}^{\;}{n_{y}{\tau_{yx} \cdot \delta}\; u_{x}\ {c}}} - {\int_{S}^{\;}{\tau_{yx}\frac{{\partial\delta}\; u_{x}}{\partial y}\ {s}}}}}{and}} & (823) \\ {{\int_{S}^{\;}{\frac{\partial\sigma_{y}}{\partial y}\delta \; u_{y}\ {s}}} = {{\int_{\; C}^{\;}{n_{y}{\sigma_{y} \cdot \delta}\; u_{y}\ {c}}} - {\int_{S}^{\;}{\sigma_{y}\frac{{\partial\delta}\; u_{y}}{\partial y}\ {s}}}}} & (824) \end{matrix}$

Substituting eqs. (821) to (824) into eq. (820) gives:

$\begin{matrix} {{\sum\limits_{i}^{\;}\; {\int_{S}^{\;}{{\left( {{\sum\limits_{j}^{\;}\; {L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}\ {s}}}} \equiv {{\frac{1}{G}{\int_{S}^{\;}{\left( {{b_{x}\delta \; u_{x}} + {b_{y}\delta \; u_{y}}} \right)\ {s}}}} + {\frac{1}{G}{\int_{C}^{\;}{\left( {{n_{x}\sigma_{x}} + {n_{y}\tau_{yx}}} \right)\delta \; u_{x}{c}}}} - {\frac{1}{G}{\int_{S}^{\;}{\left( {{{\sigma_{x}}^{\;}\frac{{\partial\delta}\; u_{x}}{\partial x}}\  + {\tau_{yx}\frac{{\partial\delta}\; u_{x}}{\partial y}}} \right){s}}}} + {\frac{1}{G}{\int_{C}^{\;}{\left( {{n_{x}\tau_{xy}} + {n_{y}\sigma_{y}}} \right)\delta \; u_{y}\ {c}}}} - {\frac{1}{G}{\int_{S}^{\;}{\left( {{\tau_{xy}\frac{{\partial\delta}\; u_{y}}{\partial x}} + {\sigma_{y}\frac{{\partial\delta}\; u_{y}}{\partial y}}} \right)\ {s}}}}}} & (825) \end{matrix}$

[Formula 367]

According to Cauchy's formula (13) and (14), the surface forces p_(x),p_(y) are given as:

p _(x) ≡n _(x)σ_(x) +n _(y)τ_(yx)  (13) (Aforementioned)

p _(y) ≡n _(x)τ_(xy) +n _(y)σ_(y)  (14) (Aforementioned)

Substituting these into (825) eq. and further utilizing the symmetricity τ_(xy)=τ_(yx) of the shear stresses of eq. (12) gives:

$\begin{matrix} {{\sum\limits_{i}^{\;}\; {\int_{S}^{\;}{{\left( {{\sum\limits_{j}^{\;}\; {L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}\ {s}}}} \equiv {{\frac{1}{G}{\int_{S}^{\;}{\left( {{b_{x}\delta \; u_{x}} + {b_{y}\delta \; u_{y}}} \right)\ {s}}}} + {\frac{1}{G}{\int_{C}^{\;}{\left( {{p_{x}\delta \; u_{x}} + {p_{y}\delta \; u_{y}}} \right){c}}}} - {\frac{1}{G}{\int_{S}^{\;}{\left\{ {{{\sigma_{x}}^{\;}\frac{{\partial\delta}\; u_{x}}{\partial x}}\  + {\tau_{xy}\left( {\frac{{\partial\delta}\; u_{x}}{\partial y} + \frac{{\partial\delta}\; u_{y}}{\partial x}} \right)} + {\sigma_{y}\frac{{\partial\delta}\; u_{y}}{\partial y}}} \right\} {s}}}}}} & (826) \end{matrix}$

The displacement-strain relationship is expressed by the following eqs. (4), (5), and (6):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 368} \right\rbrack & \; \\ {ɛ_{x} \equiv {\frac{\partial\;}{\partial x}u_{x}}} & \underset{({Aforementioned})}{(4)} \\ {ɛ_{y} \equiv {\frac{\partial\;}{\partial y}u_{y}}} & \underset{({Aforementioned})}{(5)} \\ {\gamma_{xy} \equiv {{\frac{\partial\;}{\partial y}u_{x}} + {\frac{\partial\;}{\partial x}u_{y}}}} & \underset{({Aforementioned})}{(6)} \end{matrix}$

Inverting the order of variation and differentiation according to eq. (826), and further, substituting eqs. (4) to (6) gives:

$\begin{matrix} {{\sum\limits_{i}^{\;}\; {\int_{S}^{\;}{{\left( {{\sum\limits_{j}^{\;}\; {L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}\ {s}}}} \equiv {{\frac{1}{G}{\int_{C}^{\;}{\left( {{p_{x}\delta \; u_{x}} + {p_{y}\delta \; u_{y}}} \right)\ {c}}}} + {\frac{1}{G}{\int_{S}^{\;}{\left( {{b_{x}\delta \; u_{x}} + {b_{y}\delta \; u_{y}}} \right){s}}}} - {\frac{1}{G}{\int_{S}^{\;}{\left( {{{\sigma_{x} \cdot \delta}\; ɛ_{x}}\  + {{\tau_{xy} \cdot \delta}\; \gamma_{xy}} + {{\sigma_{y} \cdot \delta}\; ɛ_{y}}} \right){s}}}}}} & (827) \end{matrix}$

Since the direct variational method of eq. (123) indicates that eq. (827) becomes zero, let eq. (827) be zero, and we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 369} \right\rbrack} & \; \\ {{\frac{1}{G}{\int_{S}^{\;}{\left( {{\sigma_{x}\delta \; ɛ_{x}}\  + {\sigma_{y}\delta \; ɛ_{xy}} + {\tau_{xy}\delta \; \gamma_{xy}}} \right){s}}}} = {\frac{1}{G}\left\lbrack {{\int_{C}^{\;}{\left( {{p_{x}\delta \; u_{x}} + {p_{y}\delta \; u_{y}}} \right)\ {c}}} + {\int_{S}^{\;}{\left( {{b_{x}\delta \; u_{x}} + {b_{y}\delta \; u_{y}}} \right)\ {s}}}} \right\rbrack}} & (828) \end{matrix}$

Eliminating the modulus of rigidity G gives:

∫_(S)(σ_(x)δε_(x)+σ_(y)δε_(y)+τ_(xy)δγ_(xy))ds=∫ _(C)(p _(x) δu _(x) +p _(y) δu _(y))dc+∫ _(S)(b _(x) δu _(x) +b _(y) δu _(y))ds  (829)

The variation δU of the strain energy is expressed by the following eq. (131):

δU≡∫ _(S)(σ_(x)δε_(x)+σ_(y)δε_(y)+τ_(xy)δγ_(xy))ds  (131) (Aforementioned)

Applying eq. (131) to the left side of the equation of eq. (829) gives the following eq. (132):

δU≡∫ _(C)(p _(x) δu _(x) +p _(y) δu _(y))dc+∫ _(S)(b _(x) δu _(x) +b _(y) δu _(y))ds  (132) (Aforementioned)

Both of eqs. (132) and (829) express nothing but the direct variational method, though their format is conventionally referred to as the principle of virtual work.

11.2.3 Variation of Strain Energy

The strain energy U is expressed by the following eq. (130):

[Formula 370]

U≡½∫_(S)(σ_(x)ε_(x)+σ_(y)ε_(y)+τ_(xy)γ_(xy))ds  (130) (Aforementioned)

The variation δU thereof is expressed by the following eq. (131):

δU≡∫ _(S)(σ_(x)δε_(x)+σ_(y)δε_(y)+τ_(xy)δγ_(xy))ds  (131) (Aforementioned)

The following description is supplementary explanation about the procedure of obtaining these. Performing variation on the strain energy U of eq. (130) gives:

$\begin{matrix} {{\delta \; U} \equiv {{\frac{1}{2}{\int_{S}^{\;}{\left( {{\delta \; \sigma_{x}ɛ_{x}} + {\sigma_{x}\delta \; ɛ_{x}}} \right)\ {s}}}} + {\frac{1}{2}{\int_{S}^{\;}{\left( {{\delta \; \sigma_{y}ɛ_{y}} + {\sigma_{y}\delta \; ɛ_{y}}} \right)\ {s}}}} + {\frac{1}{2}{\int_{S}^{\;}{\left( {{\delta \; \tau_{xy}\gamma_{xy}} + {\tau_{xy}\delta \; \gamma_{xy}}} \right)\ {s}}}}}} & (830) \end{matrix}$

In order to calculate variations of stress included in this equation, the stress-strain relationship of eqs. (7), (8), and (9) is prepared as follows:

[Formula 371]

σ_(x) =G{(μ+1)ε_(x)+(μ−1)ε_(y)}  (7) (Aforementioned)

σ_(y) =G{(μ−1)ε_(x)+(μ+1)ε_(y)}  (8) (Aforementioned)

τ_(xy) =Gγ _(xy)  (Aforementioned)

Performing variation on each gives:

δσ_(x) =G{(μ+1)δε_(x)+(μ−1)δε_(y)}  (831)

σ_(y) =G{(μ−1)δε_(x)+(μ+1)δε_(y)}  (832)

δτ_(xy) ≡Gδγ _(xy)  (833)

Substituting eqs. (831), (832), and (833) into eq. (830) gives:

$\begin{matrix} {{\delta \; U} \equiv {{\frac{1}{2}{\int_{S}^{\;}{\left\lbrack {{G\left\{ {{\left( {\mu + 1} \right)\delta \; ɛ_{x}} + {\left( {\mu - 1} \right)\delta \; ɛ_{y}}} \right\} ɛ_{x}} + {\sigma_{x}\delta \; ɛ_{x}}} \right\rbrack \ {s}}}} + {\frac{1}{2}{\int_{S}^{\;}{\left\lbrack {{G\left\{ {{\left( {\mu - 1} \right)\delta \; ɛ_{x}} + {\left( {\mu + 1} \right)\delta \; ɛ_{y}}} \right\} ɛ_{y}} + {\sigma_{y}\delta \; ɛ_{y}}} \right\rbrack \ {s}}}} + {\frac{1}{2}{\int_{S}^{\;}{\left\lbrack {{G\; \delta \; \gamma_{xy}\gamma_{xy}} + {\tau_{xy}\delta \; \gamma_{xy}}} \right\rbrack \ {s}}}}}} & (834) \end{matrix}$

Rearranging eq. (834) gives:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 372} \right\rbrack} & \; \\ {\; {{\delta \; U} \equiv {{\frac{1}{2}{\int_{S}^{\;}{\left\lbrack {{G\left\{ {{\left( {\mu + 1} \right)\; ɛ_{x}} + {\left( {\mu - 1} \right)\; ɛ_{y}}} \right\}} + \sigma_{x}} \right\rbrack \ \delta \; ɛ_{x}{s}}}} + {\frac{1}{2}{\int_{S}^{\;}{\left\lbrack {{G\left\{ {{\left( {\mu - 1} \right)ɛ_{x}} + {\left( {\mu + 1} \right)ɛ_{y}}} \right\}} + \sigma_{y}} \right\rbrack \ \delta \; ɛ_{y}{s}}}} + {\frac{1}{2}{\int_{S}^{\;}{\left\lbrack {{G\; \delta \; \gamma_{xy}} + \tau_{xy}} \right\rbrack \ \delta \; \gamma_{xy}{s}}}}}}} & (835) \end{matrix}$

Applying eqs. (7), (8), and (9) to this equation gives:

$\begin{matrix} {\; {{\delta \; U} \equiv {{\frac{1}{2}{\int_{S}^{\;}{\left( {\sigma_{x} + \sigma_{x}} \right)\delta \; ɛ_{x}\ {s}}}} + {\frac{1}{2}{\int_{S}^{\;}{\left( {\sigma_{y} + \sigma_{y}} \right)\delta \; ɛ_{y}\ {s}}}} + {\frac{1}{2}{\int_{S}^{\;}{\left( {\tau_{xy} + \tau_{xy}} \right)\delta \; \gamma_{xy}\ {s}}}}}}} & (826) \end{matrix}$

Rearranging this gives:

δU≡∫ _(S)(σ_(x)δε_(x)+σ_(y)δε_(y)+τ_(xy)δγ_(xy)  (131) (Aforementioned)

11.2.4 Energy Conservation Law

The Clapeyron's theorem, which expresses the energy conservation law, is expressed by the following eq. (134):

[Formula 373]

U=½∫_(C)(p _(x) u _(x) +p _(y) u _(y))dc+½∫_(S)(b _(x) u _(x) +b _(y) u _(y))ds  (134) (Aforementioned)

The left side of this equation represents strain energy U of the following eq. (130):

U≡½∫_(S)(σ_(x)ε_(x)+σ_(y)ε_(y)+τ_(xy)γ_(xy))ds  (130) (Aforementioned)

According to eqs. (130) and (134), we obtain:

½∫_(S)(σ_(x)ε_(x)+σ_(y)ε_(y)+τ_(xy)γ_(xy))ds=½∫_(C)(p _(x) u _(x) +p _(y) u _(y))dc+½∫_(S)(b _(x) u _(x) +b _(y) u _(y))ds  (837)

Multiplying the both sides by 2 gives:

∫_(S)(σ_(x)ε_(x)+σ_(y)ε_(y)+τ_(xy)γ_(xy))ds=∫ _(C)(p _(x) u _(x) +p _(y) u _(y))dc+∫ _(S)(b _(x) u _(x) +b _(y) u _(y))ds  (838)

This coincides with the result of eliminating the symbol δ of variation from eq. (829), as described in Chapter 7, Section (4).

11.2.5 Summary of the h-Method Element

General textbooks of the finite element method describe that the stiffness matrix of an element is configured based on the recognition of eq. (829) as expression of the principle of virtual work, but as described in Chapter 7, Section (4), actually eq. (838) expressing the energy conservation law is used. The following section describes this in detail.

11.3 Details of Equation Transformation in “Use of Conventional Finite Element” in Chapter 9 11.3.1 Shape Function

FIG. 8 referred to in Section 8.1 shows the node arrangement of a 16-nodes quadrilateral element. Though this may be used in the h-method, a node arrangement in the case of a 4-nodes quadrilateral element, which is a more typical element, is shown in FIG. 41. Shape functions thereof, N_(i) (i=1 to 4), are as follows:

[Formula 374]

N ₁≡+¼(ξ−1)(η−1)  (839)

N ₂≡−¼(ξ+1)(η−1)  (840)

N ₃≡+¼(ξ+1)(η+1)  (841)

N ₄≡−¼(ξ−1)(η+1)  (842)

These equations represent boundary internals when ξ and η are in the ranges indicated by the signs of inequality of the following expressions, and represent boundaries when ξ and η are in the ranges indicated by the equal signs of the following expressions:

−1≦ξ≦1

−1≦η≦1  (843)

11.3.2 Internal Coordinates of the Element

With use of the shape function Ni(i=1 to n{tilde over ( )}), an internal coordinate x_(j) of the element is written as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 375} \right\rbrack & \; \\ {x_{y} \equiv {\sum\limits_{i = 1}^{\overset{\_}{n}}\; {N_{i}X_{ij}}}} & \underset{({Aforementioned})}{(156)} \end{matrix}$

where X_(ij) represents a node coordinate, and n % represents the number of nodes of the element. Details of the x,y coordinates are as follows:

$\begin{matrix} {\begin{Bmatrix} x \\ y \end{Bmatrix} \equiv \begin{Bmatrix} x_{1} \\ x_{2} \end{Bmatrix} \equiv {\begin{bmatrix} N_{1} & 0 & \Lambda & N_{\overset{\_}{n}} & 0 \\ 0 & N_{1} & \Lambda & 0 & N_{\overset{\_}{n}} \end{bmatrix}\begin{Bmatrix} X_{11} \\ X_{12} \\ M \\ X_{\overset{\_}{n}1} \\ X_{\overset{\_}{n}2} \end{Bmatrix}}} & \underset{({Aforementioned})}{(157)} \end{matrix}$

Given

n∝2n%,  (158) (Aforementioned)

eq. (157) is expressed with matrices as follows:

$\begin{matrix} {\underset{2 \times 1}{\left\{ x \right\}} \equiv {\underset{2 \times n}{\lbrack N\rbrack}\underset{n \times 1}{\left\{ X \right\}}}} & \underset{({Aforementioned})}{(159)} \end{matrix}$

11.3.3 Internal Displacement of the Element

In eq. (163) of Section 8.3, a displacement u_(j) is expressed with matrices as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 376} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u \right\}} \equiv {\underset{2 \times 1}{\left\lbrack u_{A} \right\rbrack}\underset{2 \times 1}{\left\{ u_{o} \right\}}}} & \underset{({Aforementioned})}{(163)} \end{matrix}$

Here, u_(Aj) represents a boundary function, and u_(oj) represents a correction function. The boundary function u_(Aj) represents not only a displacement on the element boundary but also an internal displacement of the element, but there is no guarantee that this satisfies the differential equation. Therefore, the correction function u_(oj) is added. However, a conventional element, that is, an element of the h-method, is characterized in that an internal displacement of the element is expressed by the boundary function u_(Aj) alone. In other words, without addition of the correction function u_(oj), the displacement is expressed as follows assertively:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 378} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u \right\}} \equiv \underset{2 \times 1}{\left\{ u_{A} \right\}}} & (844) \end{matrix}$

With use of an interpolation function χ_(Ai)=(i=1˜n %), an internal displacement u_(Aj) in the element is written as:

$\begin{matrix} {u_{Aj} \equiv {\sum\limits_{i = 1}^{n\mspace{11mu} \%}{\chi_{Ai}U_{ij}}}} & \begin{matrix} {\mspace{130mu} (164)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

U_(ij) represents a nodal displacement. Here, like the method currently used for isoparametric element, the shape function N, is adopted as the interpolation function ω_(Aj). The details of the displacement are as follows:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 379} \right\rbrack} & \; \\ {{\begin{Bmatrix} u_{A\; 1} \\ u_{A\; 2} \end{Bmatrix} \equiv {\begin{bmatrix} \chi_{A\; 1} & 0 & \Lambda & \chi_{{An}\mspace{11mu} \%} & 0 \\ 0 & \chi_{A\; 1} & \Lambda & 0 & \chi_{{An}\mspace{11mu} \%} \end{bmatrix}\begin{Bmatrix} U_{11} \\ U_{12} \\ M \\ \text{?} \\ \text{?} \end{Bmatrix}}}{\text{?}\text{indicates text missing or illegible when filed}}} & \begin{matrix} {\mspace{130mu} (165)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

This is expressed with matrices as follows:

$\begin{matrix} {\underset{2 \times 1}{\left\{ u_{A} \right\}} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}} & \begin{matrix} {\mspace{130mu} (166)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

According to eqs. (844) and (166), the internal displacement of the element u_(j) of the h-method is given as:

$\begin{matrix} {\underset{2 \times 1}{\left\{ u \right\}} \equiv \underset{2 \times 1}{\left\{ u_{A} \right\}} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}} & (845) \end{matrix}$

The difference is made clear when it is compared with the following eq. (173):

$\begin{matrix} {\underset{2 \times 1}{\left\{ u \right\}} \equiv {\underset{2 \times 1}{\left\{ u_{A} \right\}} + \underset{2 \times 1}{\left\{ u_{o} \right\}}} \equiv {{\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{2 \times \lambda}{\left\lbrack \psi_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}}} & \begin{matrix} {\mspace{130mu} (173)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

11.3.4 Strain and Stress

According to eqs. (4) to (6), a strain component is expressed as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 380} \right\rbrack & \; \\ {\begin{Bmatrix} ɛ_{x} \\ ɛ_{y} \\ \gamma_{xy} \end{Bmatrix} \equiv {\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{bmatrix}\begin{Bmatrix} {\frac{\partial}{\partial x}u_{x}} \\ {\frac{\partial}{\partial x}u_{y}} \\ {\frac{\partial}{\partial y}u_{x}} \\ {\frac{\partial}{\partial y}u_{y}} \end{Bmatrix}}} & \begin{matrix} {\mspace{130mu} (192)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

This is expressed with matrices as follows:

$\begin{matrix} {\underset{3 \times 1}{\left\{ ɛ \right\}} \equiv {\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du} \right\}}}} & \begin{matrix} {\mspace{130mu} (193)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

According to eqs. (7) to (9), the stress component is given as:

$\begin{matrix} {\begin{Bmatrix} \sigma_{x} \\ \sigma_{y} \\ \tau_{xy} \end{Bmatrix} = {{G\begin{bmatrix} {\mu + 1} & {\mu - 1} & 0 \\ {\mu - 1} & {\mu + 1} & 0 \\ 0 & 0 & 1 \end{bmatrix}}\begin{Bmatrix} ɛ_{x} \\ ɛ_{y} \\ \gamma_{xy} \end{Bmatrix}}} & \begin{matrix} {\mspace{130mu} (194)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

This is expressed with matrices as follows:

$\begin{matrix} {\underset{3 \times 1}{\left\{ \sigma \right\}} = {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 1}{\left\{ ɛ \right\}}}} & \begin{matrix} {\mspace{130mu} (195)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Consequently, we obtain:

$\begin{matrix} {\underset{3 \times 1}{\left\{ \sigma \right\}} = {{G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 1}{\left\{ ɛ \right\}}} \equiv {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du} \right\}}}}} & \begin{matrix} {\mspace{130mu} (196)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

This expresses a stress component according to eqs. (10) to (12). Calculation of the coefficient matrix gives:

$\begin{matrix} \begin{matrix} {{\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}} = {\begin{bmatrix} {\mu + 1} & {\mu - 1} & 0 \\ {\mu - 1} & {\mu + 1} & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{bmatrix}}} \\ {= \begin{bmatrix} {\mu + 1} & 0 & 0 & {\mu - 1} \\ {\mu - 1} & 0 & 0 & {\mu + 1} \\ 0 & 1 & 1 & 0 \end{bmatrix}} \end{matrix} & \begin{matrix} {\mspace{130mu} (197)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

The differential coefficient of the displacement is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 381} \right\rbrack & \; \\ {\underset{4 \times 1}{\left\{ {Du} \right\}} \equiv \begin{Bmatrix} {\frac{\partial}{\partial x}u_{x}} \\ {\frac{\partial}{\partial x}u_{y}} \\ {\frac{\partial}{\partial y}u_{x}} \\ {\frac{\partial}{\partial y}u_{y}} \end{Bmatrix} \equiv \begin{Bmatrix} \underset{2 \times 1}{\frac{\partial}{\partial x}\left\{ u_{A} \right\}} \\ \underset{2 \times 1}{\frac{\partial}{\partial y}\left\{ u_{A} \right\}} \end{Bmatrix} \equiv {\begin{bmatrix} \underset{2 \times n}{\frac{\partial}{\partial x}\left\lbrack \chi_{A} \right\rbrack} \\ \underset{2 \times n}{\frac{\partial}{\partial y}\left\lbrack \chi_{A} \right\rbrack} \end{bmatrix}\underset{n \times 1}{\left\{ U \right\}}}} & (846) \end{matrix}$

The matrix on the right side of the equation is expressed as:

$\begin{matrix} \begin{matrix} {\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack} \equiv \begin{bmatrix} \underset{2 \times n}{\frac{\partial}{\partial x}\left\lbrack \chi_{A} \right\rbrack} \\ \underset{2 \times n}{\frac{\partial}{\partial y}\left\lbrack \chi_{A} \right\rbrack} \end{bmatrix}} \\ {\equiv \begin{bmatrix} {\frac{\partial}{\partial x}\chi_{A\; 1}} & 0 & \Lambda & {\frac{\partial}{\partial x}\chi_{A\; n\mspace{11mu} \%}} & 0 \\ 0 & {\frac{\partial}{\partial x}\chi_{A\; 1}} & \Lambda & 0 & {\frac{\partial}{\partial x}\chi_{A\; n\mspace{11mu} \%}} \\ {\frac{\partial}{\partial y}\chi_{A\; 1}} & 0 & \Lambda & {\frac{\partial}{\partial y}\chi_{A\; n\mspace{11mu} \%}} & 0 \\ 0 & {\frac{\partial}{\partial y}\chi_{A\; 1}} & \Lambda & 0 & {\frac{\partial}{\partial y}\chi_{A\; n\mspace{11mu} \%}} \end{bmatrix}} \end{matrix} & \begin{matrix} {\mspace{130mu} (199)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

[Formula 382]

Then, the differential coefficient of the displacement is transformed to:

$\begin{matrix} {\underset{4 \times 1}{\left\{ {Du} \right\}} \equiv {\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}} & (847) \end{matrix}$

The stress is given as:

$\begin{matrix} {\underset{3 \times 1}{\left\{ \sigma \right\}} \equiv {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du} \right\}}} \equiv {G\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}} & (848) \end{matrix}$

The dimensionless stress is given as:

$\begin{matrix} {\underset{3 \times 1}{\left\{ {\sigma/G} \right\}} \equiv {\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du} \right\}}} \equiv {\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}} & (849) \end{matrix}$

11.3.5 Strain Energy

A strain energy U is expressed by the following eq. (130):

[Formula 383]

U≡½∫_(S)(σ_(x)ε_(x)+σ_(y)ε_(y)+τ_(xy)γ_(xy))ds  (130) (Aforementioned)

This is used in the left side of the equation (837) expressing the energy conservation law. Multiplying this by 2 gives the left side of eq. (838).

Internal strain energy eng of an element is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 384} \right\rbrack & \; \\ {{eng} \equiv {\frac{1}{2}{\underset{1 \times 3}{\left\{ ɛ \right\}}}^{T}\underset{3 \times 1}{\left\{ \sigma \right\}}}} & \underset{({Aforementioned})}{(241)} \end{matrix}$

According to eqs. (193) and (196), the foregoing equation is transformed to:

$\begin{matrix} {{eng} \equiv {\frac{1}{2}G{\underset{1 \times 4}{\left\{ {Du} \right\}}}^{T}{\underset{4 \times 3}{\lbrack d\rbrack}}^{T}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times 1}{\left\{ {Du} \right\}}}} & (850) \end{matrix}$

Therefore, according to eq. (847), internal strain energy eng of an element of the h-method is given as:

$\begin{matrix} {{eng} \equiv {\frac{1}{2}G{\underset{1 \times n}{\left\{ U \right\}}}^{T}{\underset{n \times 4}{\left\lbrack {D\; \chi_{A}} \right\rbrack}}^{T}{\underset{4 \times 3}{\lbrack d\rbrack}}^{T}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}} & (851) \end{matrix}$

When this is integrated inside the element, the strain energy u of eq. (130) is obtained as:

$\begin{matrix} {{U \equiv {\frac{1}{2}{\int_{S}^{\;}{\left( {{\sigma_{x}ɛ_{x}} + {\sigma_{y}ɛ_{y}} + {\tau_{xy}\gamma_{xy}}} \right)\ {s}}}}} = {{\int_{S}^{\;}{{eng}\ {s}}} = {\frac{1}{2}G{\underset{1 \times n}{\left\{ U \right\}}}^{T}{\int_{S}^{\;}{{\underset{n \times 4}{\left\lbrack {D\; \chi_{A}} \right\rbrack}}^{T}{\underset{4 \times 3}{\lbrack d\rbrack}}^{T}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\ {s}\underset{n \times 1}{\left\{ U \right\}}}}}}} & (852) \end{matrix}$

This equation is confusing since the energy and the nodal displacement are indicated by the same sign “U”, but the “U” on the left side of the equation represents the strain energy, and two “{U}”s appearing on the right side of the equation represent the nodal displacement. The integration term of eq. (852) is defined as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 385} \right\rbrack & \; \\ {\underset{n \times n}{\left\lbrack K_{eng} \right\rbrack} \equiv {\int_{S}^{\;}{{\underset{n \times 4}{\left\lbrack {D\; \chi_{A}} \right\rbrack}}^{T}{\underset{4 \times 3}{\lbrack d\rbrack}}^{T}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\ {s}}}} & (853) \end{matrix}$

[K_(eng)] multiplied by the modulus of rigidity G represents a stiffness matrix of the h-method element, and eq. (852) indicates that the strain energy U can be calculated by:

$\begin{matrix} {U \equiv {\frac{1}{2}G{\underset{1 \times n}{\left\{ U \right\}}}^{T}\underset{n \times n}{\left\lbrack K_{eng} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}} & (854) \end{matrix}$

11.3.6 Equivalent Nodal Force

Work wrk done by an external force is given as:

[Formula 386]

wrk≡½∫_(C)(p _(x) u _(x) +p _(y) u _(y))dc++½∫_(S)(b _(x) u _(x) +b _(y) u _(y))ds  (855)

This is used in the right side of the equation (837) expressing the energy conservation law. Multiplying this by 2 gives the right side of eq. (838). Let terms p_(x), p_(y) of the surface force integrated on the boundary and divided to nodes be F_(p), and let terms b_(x), b_(y) of the body force integrated inside the element and divided to nodes be F_(b). Then, an equivalent nodal force per unit thickness is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 387} \right\rbrack & \; \\ {\underset{n \times 1}{\left\{ F \right\}} \equiv {\underset{n \times 1}{\left\{ F_{p} \right\}} + \underset{n \times 1}{\left\{ F_{b} \right\}}}} & (856) \end{matrix}$

Using equivalent nodal forces F_(p),F_(b), we define work done by surface force as:

$\begin{matrix} {{\frac{1}{2}{\int_{C}^{\;}{\left( {{p_{x}u_{x}} + {p_{y}u_{y}}} \right)\ {c}}}} \equiv {\frac{1}{2}{\underset{1 \times n}{\left\{ U \right\}}}^{T}\underset{n \times 1}{\left\{ F_{p} \right\}}}} & (857) \end{matrix}$

We define work done by body force as:

$\begin{matrix} {{\frac{1}{2}{\int_{S}^{\;}{\left( {{b_{x}u_{x}} + {b_{y}u_{y}}} \right)\ {s}}}} \equiv {\frac{1}{2}{\underset{1 \times n}{\left\{ U \right\}}}^{T}\underset{n \times 1}{\left\{ F_{b} \right\}}}} & (858) \end{matrix}$

These are combined and the work wrk done by external force expressed by eq. (855) is expressed as:

$\begin{matrix} {{wrk} \equiv {{\frac{1}{2}{\int_{C}^{\;}{\left( {{p_{x}u_{x}} + {p_{y}u_{y}}} \right)\ {c}}}} + {\frac{1}{2}{\int_{S}^{\;}{\left( {{b_{x}u_{x}} + {b_{y}u_{y}}} \right)\ {s}}}}} \equiv {{\frac{1}{2}{\underset{1 \times n}{\left\{ U \right\}}}^{T}\underset{n \times 1}{\left\{ F_{p} \right\}}} + {\frac{1}{2}{\underset{1 \times n}{\left\{ U \right\}}}^{T}\underset{n \times 1}{\left\{ F_{b} \right\}}}} \equiv {\frac{1}{2}{\underset{1 \times n}{\left\{ U \right\}}}^{T}\underset{n \times 1}{\left\{ F \right\}}}} & (859) \end{matrix}$

11.3.7 Element Stiffness Matrix of the h-Method

According to the energy conservation law of eq. (837), eq. (854) and eq. (859) are equated, which is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 388} \right\rbrack & \; \\ {{\frac{1}{2}G{\underset{1 \times n}{\left\{ U \right\}}}^{T}\underset{n \times n}{\left\lbrack K_{eng} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} = {\frac{1}{2}{\underset{1 \times n}{\left\{ U \right\}}}^{T}\underset{n \times 1}{\left\{ F \right\}}}} & (860) \end{matrix}$

In order that this equation should be established with respect to an arbitrary nodal displacement U, the following has to be given:

$\begin{matrix} {{G\underset{n \times n}{\left\lbrack K_{eng} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} = \underset{n \times 1}{\left\{ F \right\}}} & (861) \end{matrix}$

As this equation is an expression per unit thickness, this equation is multiplied by a plate thickness h in order to deal with a plane stress state, and is given as:

$\begin{matrix} {{{Gh}\underset{n \times n}{\left\lbrack K_{eng} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} = \underset{n \times 1}{\left\{ {Fh} \right\}}} & (862) \end{matrix}$

Dividing the both sides by a modulus of rigidity G gives:

$\begin{matrix} {{\underset{n \times n}{\left\lbrack K_{eng} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} = \underset{n \times 1}{\left\{ {F/G} \right\}}} & (863) \end{matrix}$

To recognize this is useful. We define an element stiffness matrix [K_(elm)] per unit thickness as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 389} \right\rbrack & \; \\ {\underset{n \times n}{\left\lbrack K_{elm} \right\rbrack} \equiv {G\underset{n \times n}{\left\lbrack K_{eng} \right\rbrack}}} & (864) \end{matrix}$

Then, eq. (861) is transformed to:

$\begin{matrix} {{\underset{n \times n}{\left\lbrack K_{elm} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} = \underset{n \times 1}{\left\{ F \right\}}} & (865) \end{matrix}$

This expresses equivalence between the nodal displacement U and the equivalent nodal force F in terms of energy.

We divide an object as an analysis object into an enormous number of elements, create the equations (865) with regard to the elements, respectively, and compose an equation of the whole system, while being careful about shared nodes. Then, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 390} \right\rbrack & \; \\ {{\underset{N \times N}{\lbrack K\rbrack}\underset{N \times 1}{\left\{ U \right\}}} = \underset{N \times 1}{\left\{ F \right\}}} & (866) \end{matrix}$

where N represents the total number of degrees of freedom of the whole system. The total degrees of freedom N is represented by “n” in eq. (434). A method for solution by recognizing this as equilibrium of force is the linear elastic finite element method by the h-method, that is, the conventional method.

Closely studying the procedure for obtaining eq. (865), we can see that no variation operation is used anywhere, and no virtual displacement appears. In many textbooks, it is explained as if eq. (829) is calculated whereby eq. (865) is obtained, but actually this explanation is not correct. Actually the energy conservation law of eqs. (837), (838) is used. Therefore, it is rational to understand that eqs. (866), (434) express equivalence in terms of energy, rather than to understand that they express equilibrium of force.

Therefore, as shown in Section 9.2, it is not necessary to be bound by a self-adjoint boundary condition, but eqs. (866), (434) may be solved under a non-self-adjoint boundary condition.

11.3.8 Characteristics of Finite Element

Regarding a new finite element, the equivalent nodal force F per unit thickness is expressed by eq. (214), which is as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 391} \right\rbrack & \; \\ {\underset{n \times 1}{\left\{ F \right\}} \equiv {G\left( {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o} \right\}}}} \right)}} & \underset{({Aforementioned})}{(214)} \end{matrix}$

The dual equivalent nodal force F* per unit thickness is expressed by eq. (233), which is as follows:

$\begin{matrix} {\underset{n \times 1}{\left\{ F^{*} \right\}} \equiv {G\left( {{\underset{n \times n}{\left\lbrack K_{U} \right\rbrack}\underset{n \times 1}{\left\{ U^{*} \right\}}} + {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack}\underset{\lambda \times 1}{\left\{ c_{o}^{*} \right\}}}} \right)}} & \underset{({Aforementioned})}{(233)} \end{matrix}$

Here, details the matrix part is expressed by eqs. (212), (213) as follows:

$\begin{matrix} {\underset{n \times n}{\left\lbrack K_{U} \right\rbrack} \equiv {\int_{C}^{\;}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack^{T}}\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\ {c}}}} & \underset{({Aforementioned})}{(212)} \\ {\underset{n \times \lambda}{\left\lbrack K_{o} \right\rbrack} \equiv {\int_{C}^{\;}{\underset{n \times 2}{\left\lbrack \chi_{A} \right\rbrack^{T}}\underset{2 \times 3}{\lbrack T\rbrack}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times \lambda}{\left\lbrack {D\; \psi_{o}} \right\rbrack}\ {c}}}} & \underset{({Aforementioned})}{(213)} \end{matrix}$

These are obtained by integration of a surface force on an element boundary.

In contrast, in the case of the conventional finite element method, that is, in the case of an element of the h-method, the equivalent nodal force F per unit thickness is defined by eq. (861) as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 392} \right\rbrack & \; \\ {\underset{n \times 1}{\left\{ F \right\}} = {G\underset{n \times n}{\left\lbrack K_{eng} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}} & \underset{({Aforementioned})}{(861)} \end{matrix}$

Here, details the matrix part is expressed by eq. (853) as follows:

$\begin{matrix} {\underset{n \times n}{\left\lbrack K_{eng} \right\rbrack} \equiv {\int_{S}^{\;}{\underset{n \times 4}{\left\lbrack {D\; \chi_{A}} \right\rbrack^{T}}\underset{4 \times 3}{\lbrack d\rbrack^{T}}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\ {s}}}} & \underset{({Aforementioned})}{(853)} \end{matrix}$

This is obtained by equating the internal strain energy of the element and work done by the equivalent nodal force.

While the stiffness matrix in the case of the new finite element is composed of two matrices of [K_(U)], [K_(O)], the stiffness matrix in the case of the h-method element is composed of one matrix of [K_(eng)]. The conventional and new stiffness matrices are completely different in their characteristics as seen in details thereof, that is, the calculation methods of eqs. (212), (213), and eq. (853), and we can see that this does not mean that the two coincide if effects by a correction function, that is, [K_(O)], are ignored.

11.3.9 Stiffness Matrix of Rectangular Element

Regarding a rectangular element having node arrangement as shown in FIG. 41, let coordinates of nodes 1, 2, 3, and 4 be (−a,−b), (a,−b), (a,b), and (−a,b), respectively, and let an aspect ratio κ thereof be:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 393} \right\rbrack & \; \\ {\kappa \equiv \frac{b}{a}} & (867) \end{matrix}$

With respect to this rectangular element, calculation of an integration term expressed by eq. (853),

$\begin{matrix} {\underset{n \times n}{\left\lbrack K_{eng} \right\rbrack} \equiv {\int_{S}^{\;}{\underset{n \times 4}{\left\lbrack {D\; \chi_{A}} \right\rbrack^{T}}\underset{4 \times 3}{\lbrack d\rbrack^{T}}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}\ {s}}}} & \underset{({Aforementioned})}{(853)} \end{matrix}$

is performed.

The internal coordinates of the element are expressed by (159):

$\begin{matrix} {\underset{2 \times 1}{\left\{ x \right\}} \equiv {\underset{2 \times n}{\lbrack N\rbrack}\underset{n \times 1}{\left\{ X \right\}}}} & \underset{({Aforementioned})}{(159)} \end{matrix}$

As a result, internal coordinates x_(j) of the 4-nodes element shown in 41 are given as:

[Formula 394]

x ₁ ≡aξ

x ₂ ≡bη  (156) (Aforementioned)

Therefore, a Jacobian (Jacobian) J is given as:

J≡ab  (868)

And an area fragment ds of the integration term of eq. (199) is given as:

ds≡Jdξdη  (869)

The integration variable can be converted, and we obtain:

$\begin{matrix} {\underset{n \times n}{\left\lbrack K_{eng} \right\rbrack} \equiv {\int_{\; {- 1}}^{1}{\int_{- 1}^{1}{\underset{n \times 4}{\left\lbrack {D\; \chi_{A}} \right\rbrack^{T}}{\underset{4 \times 3}{\lbrack d\rbrack}}^{T}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}\underset{4 \times n}{\left\lbrack {D\; \chi_{A}} \right\rbrack}J\ {\xi}\ {\eta}}}}} & (870) \end{matrix}$

The internal displacement u_(j) of the element is expressed by eq. (845) as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 395} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ u \right\}} \equiv \underset{2 \times 1}{\left\{ u_{A} \right\}} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}} & \underset{({Aforementioned})}{(845)} \end{matrix}$

Details of the same are given as:

$\begin{matrix} {\begin{Bmatrix} u_{1} \\ u_{2} \end{Bmatrix} \equiv \begin{Bmatrix} u_{A\; 1} \\ u_{A\; 2} \end{Bmatrix} \equiv {\begin{bmatrix} \chi_{A\; 1} & 0 & \Lambda & \chi_{A\; 4} & 0 \\ 0 & \chi_{A\; 1} & \Lambda & 0 & \chi_{A\; 4} \end{bmatrix}\begin{Bmatrix} U_{11} \\ U_{12} \\ M \\ U_{41} \\ U_{42} \end{Bmatrix}}} & (871) \end{matrix}$

Here, like the method currently used for isoparametric element, the shape function N_(i) is adopted as the interpolation function χ_(Ai). U_(ij) represents a nodal displacement, and depending on the arrangement thereof, the arrangement of matrices of various types is settled. According to eq. (192), (193), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 396} \right\rbrack & \; \\ {\underset{3 \times 4}{\lbrack d\rbrack} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{bmatrix}} & (872) \end{matrix}$

According to eqs. (194), (195), we obtain:

$\begin{matrix} {\underset{3 \times 3}{\lbrack\mu\rbrack} = \begin{bmatrix} {\mu + 1} & {\mu - 1} & 0 \\ {\mu - 1} & {\mu + 1} & 0 \\ 0 & 0 & 1 \end{bmatrix}} & (873) \end{matrix}$

As a result, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 397} \right\rbrack & \; \\ {{\underset{4 \times 3}{\left\lbrack d^{\Gamma} \right\rbrack}\underset{3 \times 3}{\lbrack\mu\rbrack}\underset{3 \times 4}{\lbrack d\rbrack}} \equiv \begin{bmatrix} {\mu + 1} & 0 & 0 & {\mu - 1} \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ {\mu - 1} & 0 & 0 & {\mu + 1} \end{bmatrix}} & (874) \end{matrix}$

With regard to a differential coefficient of the displacement, we obtain, according to eq. (199), as follows:

$\begin{matrix} {\underset{4 \times 8}{\left\lbrack D_{\chi_{A}} \right\rbrack} \equiv \begin{bmatrix} \frac{\eta - 1}{4a} & 0 & {- \frac{\eta - 1}{4a}} & 0 & \frac{\eta + 1}{4a} & 0 & {- \frac{\eta - 1}{4a}} & 0 \\ 0 & \frac{\eta - 1}{4a} & 0 & {- \frac{\eta - 1}{4a}} & 0 & \frac{\eta + 1}{4a} & 0 & {- \frac{\eta + 1}{4a}} \\ \frac{\zeta - 1}{4b} & 0 & {- \frac{\zeta + 1}{4b}} & 0 & \frac{\zeta + 1}{4b} & 0 & {- \frac{\zeta - 1}{4b}} & 0 \\ 0 & \frac{\zeta - 1}{4b} & 0 & {- \frac{\zeta + 1}{4b}} & 0 & \frac{\zeta + 1}{4b} & 0 & {- \frac{\zeta - 1}{4b}} \end{bmatrix}} & (875) \end{matrix}$

As a result of the foregoing operation, eq. (870) can be calculated, and we obtain:

$\begin{matrix} {\underset{8 \times 8}{\left\lbrack K_{eng} \right\rbrack} \equiv {\quad\begin{bmatrix} A_{1} & B_{1} & A_{3} & {- B_{2}} & A_{7} & {- B_{1}} & A_{5} & B_{2} \\ B_{1} & A_{2} & B_{2} & A_{4} & {- B_{1}} & A_{8} & {- B_{2}} & A_{6} \\ A_{3} & B_{2} & A_{1} & {- B_{1}} & A_{5} & {- B_{2}} & A_{7} & B_{1} \\ {- B_{2}} & A_{4} & {- B_{1}} & A_{2} & B_{2} & A_{6} & B_{1} & A_{8} \\ A_{7} & {- B_{1}} & A_{5} & B_{2} & A_{1} & B_{1} & A_{3} & {- B_{2}} \\ {- B_{1}} & A_{8} & {- B_{2}} & A_{6} & B_{1} & A_{2} & B_{2} & A_{4} \\ A_{5} & {- B_{2}} & A_{7} & B_{1} & A_{3} & B_{2} & A_{1} & {- B_{1}} \\ B_{2} & A_{6} & B_{1} & A_{8} & {- B_{2}} & A_{4\;} & {- B_{1}} & A_{2} \end{bmatrix}}} & (876) \end{matrix}$

Here, the following are given:

$\begin{matrix} {C_{1} \equiv {{\left( {1 + \mu} \right)\kappa} + \frac{1}{\kappa}}} & (877) \\ {C_{2} \equiv {\kappa + {\left( {1 + \mu} \right)\frac{1}{\kappa}}}} & (878) \\ {C_{3} \equiv {{{- 2}\left( {1 + \mu} \right)\kappa} + \frac{1}{\kappa}}} & (879) \\ {C_{4} \equiv {{{- 2}\kappa} + {\left( {1 + \mu} \right)\frac{1}{\kappa}}}} & (880) \\ {C_{5} \equiv {{\left( {1 + \mu} \right)\kappa} - \frac{2}{\kappa}}} & (881) \\ {C_{6} \equiv {\kappa - {\left( {1 + \mu} \right)\frac{2}{\kappa}}}} & (882) \end{matrix}$

Then, we obtain:

$\begin{matrix} {A_{1} \equiv \frac{C_{1}}{3}} & (883) \\ {A_{2} \equiv \frac{C_{2}}{3}} & (884) \\ {A_{3} \equiv \frac{C_{3}}{6}} & (885) \\ {A_{4} \equiv \frac{C_{4}}{6}} & (886) \\ {A_{5} \equiv \frac{C_{5}}{6}} & (887) \\ {A_{6} \equiv \frac{C_{6}}{6}} & (888) \\ {A_{7} \equiv {- \frac{C_{1}}{6}}} & (889) \\ {A_{8} \equiv {- \frac{C_{2}}{6}}} & (890) \end{matrix}$

Further, we obtain:

$\begin{matrix} {B_{1} \equiv \frac{\mu}{4}} & (891) \\ {B_{2} \equiv {\frac{1}{4}\left( {2 - \mu} \right)}} & (892) \end{matrix}$

It should be noted that the constant μ is given by eq. (3) as follows, with the Poisson's ratio being given as v:

$\begin{matrix} {{\mu \equiv {\frac{1 + v}{1 - v}\left( {{Plane}\mspace{14mu} {Stress}} \right)}},{\mu \equiv {\frac{1}{1 - {2v}}\left( {{Plane}\mspace{14mu} {Strain}} \right)}}} & \underset{({Aforementioned})}{(3)} \end{matrix}$

Regarding the element stiffness matrix [K_(elm)] per unit thickness, substituting eq. (876) into eq. (864) allows both of the plane stress state and the plane strain state to be given as:

$\begin{matrix} {\underset{8 \times 8}{\left\lbrack K_{elm} \right\rbrack} \equiv {G\underset{8 \times 8}{\left\lbrack K_{eng} \right\rbrack}}} & (893) \end{matrix}$

Incidentally, textbooks about the finite element method usually define the element stiffness matrix [K_(elm)] in a format of a matrix of eq. (876) multiplied by the coefficient c*, as follows:

$\begin{matrix} {\underset{8 \times 8}{\left\lbrack K_{elm} \right\rbrack} \equiv {c^{*}{\quad\begin{bmatrix} A_{1}^{*} & B_{1}^{*} & A_{3}^{*} & {- B_{2}^{*}} & A_{7}^{*} & {- B_{1}^{*}} & A_{5}^{*} & B_{2}^{*} \\ B_{1}^{*} & A_{2}^{*} & B_{2}^{*} & A_{4}^{*} & {- B_{1}^{*}} & A_{8}^{*} & {- B_{2}^{*}} & A_{6}^{*} \\ A_{3}^{*} & B_{2}^{*} & A_{1}^{*} & {- B_{1}^{*}} & A_{5}^{*} & {- B_{2}^{*}} & A_{7}^{*} & B_{1}^{*} \\ {- B_{2}^{*}} & A_{4}^{*} & {- B_{1}^{*}} & A_{2}^{*} & B_{2}^{*} & A_{6}^{*} & B_{1}^{*} & A_{8}^{*} \\ A_{7}^{*} & {- B_{1}^{*}} & A_{5}^{*} & B_{2}^{*} & A_{1}^{*} & B_{1}^{*} & A_{3}^{*} & {- B_{2}^{*}} \\ {- B_{1}^{*}} & A_{8}^{*} & {- B_{2}^{*}} & A_{6}^{*} & B_{1}^{*} & A_{2}^{*} & B_{2}^{*} & A_{4}^{*} \\ A_{5}^{*} & {- B_{2}^{*}} & A_{7}^{*} & B_{1}^{*} & A_{3}^{*} & B_{2}^{*} & A_{1}^{*} & {- B_{1}^{*}} \\ B_{2}^{*} & A_{6}^{*} & B_{1}^{*} & A_{8}^{*} & {- B_{2}^{*}} & A_{4\;}^{*} & {- B_{1}^{*}} & A_{2} \end{bmatrix}}}} & (894) \end{matrix}$

In the case where this equation is used, it is necessary to rewrite the matrix element to make the matrix element different in the case of the plane stress state and in the case of the plane strain state, and this is slightly complicating. However, as it is also important to show the equation seen often in textbooks and know correspondence between the equation and eq. (893), the matrix elements of the both are shown below. We can see that eq. (893), which does not require distinguishing the two states, can be handled easily.

In the plane stress state, let a Young's modulus (Young's modulus) be E, and we obtain a coefficient c* of eq. (894) as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 403} \right\rbrack & \; \\ {c^{*} \equiv \frac{E}{12\left( {1 - v^{2}} \right)}} & (895) \end{matrix}$

Given

$\begin{matrix} {C_{1}^{*} \equiv {{2\kappa} + \frac{1 - v}{\kappa}}} & (896) \\ {C_{2}^{*} \equiv {{\left( {1 - v} \right)\kappa} + \frac{2}{\kappa}}} & (897) \\ {C_{3}^{*} \equiv {{{- 4}\kappa} + \frac{1 - v}{\kappa}}} & (898) \\ {C_{4}^{*} \equiv {{{- 4}{\kappa \left( {1 - v} \right)}} + \frac{2}{\kappa}}} & (899) \\ {C_{5}^{*} \equiv {{2\kappa} - \frac{2\left( {1 - v} \right)}{\kappa}}} & (900) \\ {{C_{6}^{*} \equiv {{\left( {1 - v} \right)\kappa} - \frac{4}{\kappa}}},} & (901) \end{matrix}$

we obtain:

[Formula 404]

A ₁*≡2C ₁*  (902)

A ₂*≡2C ₂*  (903)

A ₃ *≡C ₃*  (904)

A ₄ *≡C ₄*  (905)

A ₅ *≡C ₅*  (906)

A ₆ *≡C ₆*  (907)

A ₇ *≡−C ₁*  (908)

A ₈ *≡−C ₁*  (909)

Further, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 405} \right\rbrack & \; \\ {B_{1}^{*} \equiv {\frac{3}{2}\left( {1 + v} \right)}} & (910) \\ {B_{2}^{*} \equiv {\frac{3}{2}\left( {1 - {3v}} \right)}} & (911) \end{matrix}$

It should be noted that an element stiffness matrix [K_(elm)] obtained by substituting the following equation (912) regarding a material constant and the following value (913) of eq. (3) in the plane stress state into eq. (893) coincides with eq. (894) in the plane stress state.

$\begin{matrix} {G = \frac{E}{2\left( {1 + v} \right)}} & (912) \\ {\mu \equiv \frac{1 + v}{1 - v}} & (913) \end{matrix}$

Similarly, in the plane strain state, let a Young's modulus be E, and we obtain a coefficient c* of eq. (894) as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 406} \right\rbrack & \; \\ {c^{*} \equiv \frac{E\left( {1 - v} \right)}{12\left( {1 + v} \right)\left( {1 - {2v}} \right)}} & (914) \end{matrix}$

Given

$\begin{matrix} {C_{1}^{*} \equiv {\kappa + {\frac{1 - {2v}}{2\left( {1 - v} \right)}\frac{1}{\kappa}}}} & (915) \\ {C_{2}^{*} \equiv {{\frac{1 - {2v}}{2\left( {1 - v} \right)}\kappa} + \frac{1}{\kappa}}} & (916) \\ {C_{3}^{*} \equiv {{{- 4}\kappa} + {\frac{1 - {2v}}{\left( {1 - v} \right)}\frac{1}{\kappa}}}} & (917) \\ {C_{4}^{*} \equiv {{{- \frac{2\left( {1 - {2v}} \right)}{\left( {1 - v} \right)}}\kappa} + \frac{2}{\kappa}}} & (918) \\ {C_{5}^{*} \equiv {{2\kappa} - {\frac{2\left( {1 - {2v}} \right)}{\left( {1 - v} \right)}\frac{1}{\kappa}}}} & (919) \\ {{C_{6}^{*} \equiv {{\frac{1 - {2v}}{\left( {1 - v} \right)}\kappa} - \frac{4}{\kappa}}},} & (920) \end{matrix}$

we obtain:

[Formula 407]

A ₁*≡4C ₁*  (921)

A ₂*≡4C ₂*  (922)

A ₃ *≡C ₃*  (923)

A ₄ *≡C ₄*  (924)

A ₅ *≡C ₅*  (925)

A ₆ *≡C ₆*  (926)

A ₇*≡−2C ₁*  (927)

A ₈*≡−2C ₂*  (928)

Further, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 408} \right\rbrack & \; \\ {B_{1}^{*} \equiv {\frac{3}{2}\left( \frac{1}{1 - v} \right)}} & (929) \\ {B_{2}^{*} \equiv {\frac{3}{2}\left( \frac{1 - {4v}}{1 - v} \right)}} & (930) \end{matrix}$

It should be noted that an element stiffness matrix [K_(elm)] obtained by substituting the following equation (912) regarding a material constant and the following value (931) of eq. (3) in the plane strain state into eq. (893) coincides with eq. (894) in the plane strain state.

$\begin{matrix} {G = \frac{E}{2\left( {1 + v} \right)}} & \underset{({Aforementioned})}{(912)} \\ {\mu \equiv \frac{1}{1 - {2v}}} & (931) \end{matrix}$

11.3.10 Supplementary Description about Underdetermined System in which a Plurality of Solutions Exist

In Section 9.4, the number of homogeneous solutions n_(o) is given as:

[Formula 409]

n _(o) ≡n _(v) −n  (453) (Aforementioned)

where n represents the number of degrees of freedom in the whole structure, and n_(v) represents the number of unknown values. This assumes a case where either the rank of the matrix [K_(v)] in either the following equation (447) or the following equation (452) is equal to n, and the following equation (932) is satisfied:

$\begin{matrix} {{\underset{n \times n_{v}}{\left\lbrack K_{v} \right\rbrack}\underset{n_{v} \times 1}{\left\{ s_{v} \right\}}} = {{- \underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack}}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}}} & \underset{({Aforementioned})}{(447)} \\ {{\underset{n \times n_{v}}{\left\lbrack K_{v} \right\rbrack}\underset{n_{v} \times 1}{\left\{ s_{v} \right\}}} = {{{- \underset{n \times n_{b}}{\left\lbrack K_{b} \right\rbrack}}\underset{n_{b} \times 1}{\left\{ s_{b} \right\}}} + \underset{n \times 1}{\left\{ F_{g} \right\}}}} & \underset{({Aforementioned})}{(452)} \\ {{{rank}\left\lbrack K_{v} \right\}} = n} & (932) \end{matrix}$

In the case where the rank of the matrix [K_(v)] is smaller than n, we obtain the following based on the linear algebra:

n _(o) ≡n _(v)−rank[K _(v)]  (933)

In this case, eq. (933) may be adopted in place of eq. (453).

11.3.11 Supplementary Description about the Least-Squares Method

The following is supplementary explanation about the calculation method of the variational principle shown in Section 9.5. First of all, similarly to eq. (251), the nodal displacement U is divided in to a known part U_(b) and an unknown part U_(v), and is defined as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 410} \right\rbrack & \; \\ {\underset{n \times 1}{\left\{ U \right\}} \equiv \begin{bmatrix} \underset{n_{v_{b}} \times 1}{\left\{ U_{b} \right\}} \\ \underset{n_{U_{b}} \times 1}{\left\{ U_{v} \right\}} \end{bmatrix}} & (934) \end{matrix}$

where n represents the total number of degrees of freedom in the whole structure. Similarly, the nodal external force F is divided in to a known part F_(b) and an unknown part F_(v), and is defined as:

$\begin{matrix} {\underset{n \times 1}{\left\{ F \right\}} \equiv \begin{bmatrix} \underset{n_{F_{b}} \times 1}{\left\{ F_{b} \right\}} \\ \underset{n_{F_{v}} \times 1}{\left\{ F_{v} \right\}} \end{bmatrix}} & (935) \end{matrix}$

This is identical to the contents described in section 8.11, except that n is regarded as the total number of degrees of freedom when the whole structure is divided into mesh and is expressed by the h-method element.

Let a mode coefficient with respect to a homogeneous solution be {a_(o)}, and we obtain a solution {s_(v)} expressed by the following eq. (455):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 411} \right\rbrack & \; \\ {\underset{n_{v} \times 1}{\left\{ s_{v} \right\}} = {\underset{n_{v} \times 1}{\left\{ s_{p} \right\}} + {\underset{n_{v} \times n_{o}}{\left\lbrack \psi_{o} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}}} & \underset{({Aforementioned})}{(455)} \end{matrix}$

The node unknown part s_(v) is expressed by eq. (444) as follows:

$\begin{matrix} {\underset{n_{v} \times 1}{\left\{ s_{v} \right\}} \equiv \begin{bmatrix} \underset{n_{U_{v}} \times 1}{\left\{ U_{v} \right\}} \\ \underset{n_{F_{v}} \times 1}{\left\{ F_{v} \right\}} \end{bmatrix}} & \underset{({Aforementioned})}{(444)} \end{matrix}$

The particular solution {s_(p)} and the matrix [ψ_(o)] of eq. (455) is divided into two, i.e., the upper part and the lower part, and are defined as follows:

$\begin{matrix} {{\begin{Bmatrix} \underset{n_{U_{v}} \times 1}{\left\{ s_{pU} \right\}} \\ \underset{n_{F_{v}} \times 1}{\left\{ s_{pF} \right\}} \end{Bmatrix} \equiv \underset{n_{v} \times 1}{\left\{ s_{p} \right\}}}{and}} & (936) \\ \left\lbrack {{Formula}\mspace{14mu} 412} \right\rbrack & \; \\ {\begin{bmatrix} \underset{n_{U_{v}} \times n_{o}}{\left\lbrack \psi_{oU} \right\rbrack} \\ \underset{n_{F_{v}} \times n_{o}}{\left\lbrack \psi_{oF} \right\rbrack} \end{bmatrix} \equiv \underset{n_{\;_{v}} \times n_{o}}{\left\lbrack \psi_{o} \right\rbrack}} & (937) \end{matrix}$

Then, we obtain the unknown part U_(v) of the nodal displacement U given as:

$\begin{matrix} {\underset{n_{U_{v}} \times 1}{\left\{ U_{v} \right\}} = {\underset{n_{U_{v}} \times 1}{\left\{ s_{pU} \right\}} + {\underset{n_{U_{v}} \times n_{o}}{\left\lbrack \psi_{oU} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}}} & (938) \end{matrix}$

Also, we obtain the unknown part F of the nodal external force F given as:

$\begin{matrix} {\underset{n_{F_{v}} \times 1}{\left\{ F_{v} \right\}} = {\underset{n_{F_{v}} \times 1}{\left\{ s_{pF} \right\}} + {\underset{n_{F_{v}} \times n_{o}}{\left\lbrack \psi_{oF} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}}} & (939) \end{matrix}$

Both of the unknown parts U_(v) and F_(v) are expressed with the mode coefficient {a_(o)}, which means that the problem is solved completely. In other words, the problem is solved with respect to an arbitrary {a_(o)}.

However, this solution merely represents the equivalence in terms of energy as described in Section 11.3.7, and there is no guarantee that the differential equations are satisfied inside the element and in the whole structure. Therefore, by using the variational principle of eq. (122) and the least squares method so as to obtain a solution that approximately satisfies the differential equations in the whole structure system, {a_(o)} is determined by the variational principle, and thereafter, the unknown displacement is determined by eq. (938), and the unknown external force is determined by eq. (939). Since it can be considered that the solution satisfying the approximately differential equations can be expressed in the vicinities of {a_(o)} determined by the variational principle, a designer is allowed to proceed designing work, while confirming the state of feasible deformation, stress distribution, error distribution, and the like by operating a slider.

According to eqs. (934), (938), the nodal displacement U is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 413} \right\rbrack & \; \\ {{\underset{n \times 1}{\left\{ U \right\}} \equiv \begin{Bmatrix} \underset{n_{U_{b}} \times 1}{\left\{ U_{b} \right\}} \\ \underset{n_{U_{v}} \times 1}{\left\{ U_{v} \right\}} \end{Bmatrix}} = {\begin{Bmatrix} \underset{n_{U_{b}} \times 1}{\left\{ U_{b} \right\}} \\ \underset{n_{U_{v}} \times 1}{\left\{ s_{pU} \right\}} \end{Bmatrix} + \begin{Bmatrix} \underset{n_{U_{b}} \times 1}{\left\{ 0 \right\}} \\ {\underset{n_{U_{v} \times n_{o}}}{\left\lbrack \psi_{oU} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}} \end{Bmatrix}}} & (940) \end{matrix}$

Here, by defining

$\begin{matrix} {{\underset{n \times 1}{\left\{ U_{p} \right\}} \equiv \begin{Bmatrix} \underset{n_{U_{b}} \times 1}{\left\{ U_{b} \right\}} \\ \underset{n_{U_{v}} \times 1}{\left\{ s_{pU} \right\}} \end{Bmatrix}}{and}} & (941) \\ {{\underset{n \times n_{o}}{\left\lbrack \psi_{U} \right\rbrack} \equiv \begin{bmatrix} \underset{n_{U_{b}} \times n_{o}}{\lbrack 0\rbrack} \\ \underset{n_{U_{v} \times n_{o}}}{\left\lbrack \psi_{oU} \right\rbrack} \end{bmatrix}},} & (942) \end{matrix}$

we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 414} \right\rbrack & \; \\ {\underset{n \times 1}{\left\{ U \right\}} = {\underset{n \times 1}{\left\{ U_{p} \right\}} + {\underset{n \times n_{o}}{\left\lbrack \psi_{U} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}}} & (944) \end{matrix}$

Similarly, according to eqs. (935), (939), the nodal external force F is given as:

$\begin{matrix} {{\underset{n \times 1}{\left\{ F \right\}} \equiv \begin{Bmatrix} \underset{n_{F_{b}} \times 1}{\left\{ F_{b} \right\}} \\ \underset{n_{F_{v}} \times 1}{\left\{ F_{v} \right\}} \end{Bmatrix}} = {\begin{Bmatrix} \underset{n_{F_{b}} \times 1}{\left\{ F_{b} \right\}} \\ \underset{n_{F_{v}} \times 1}{\left\{ s_{p\; F} \right\}} \end{Bmatrix} + \begin{Bmatrix} \underset{n_{F_{b}} \times 1}{\left\{ 0 \right\}} \\ {\underset{n_{F_{v}} \times n_{o}}{\left\lbrack \psi_{oF} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}} \end{Bmatrix}}} & (944) \end{matrix}$

Here, defining

$\begin{matrix} {{\underset{n \times 1}{\left\{ F_{p} \right\}} \equiv \begin{Bmatrix} \underset{n_{F_{b}} \times 1}{\left\{ F_{b} \right\}} \\ \underset{n_{F_{v}} \times 1}{\left\{ s_{p\; F} \right\}} \end{Bmatrix}}{and}} & (945) \\ \left\lbrack {{Formula}\mspace{14mu} 415} \right\rbrack & \; \\ {{\underset{n \times n_{o}}{\left\lbrack \psi_{F} \right\rbrack} \equiv \begin{bmatrix} \underset{n_{F_{b}} \times n_{o}}{\lbrack 0\rbrack} \\ \underset{n_{F_{v}} \times n_{o}}{\left\lbrack \psi_{oF} \right\rbrack} \end{bmatrix}},} & (946) \end{matrix}$

we obtain:

$\begin{matrix} {\underset{n \times 1}{\left\{ F \right\}} = {\underset{n \times 1}{\left\{ F_{p} \right\}} + {\underset{n \times n_{o}}{\left\lbrack \psi_{F} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}}} & (947) \end{matrix}$

On the other hand, in the h-method element, the internal displacement of the element is expressed by eq. (166) as follows:

$\begin{matrix} {{\underset{2 \times 1}{\left\{ u_{A} \right\}} \equiv {\underset{2 \times n}{\left\lbrack \chi_{A} \right\rbrack}\underset{n \times 1}{\left\{ U \right\}}}}({Aforementioned})} & (166) \end{matrix}$

where n represents the number of degrees of the element. Since the same sign “n” used for different items makes the equation complicating, the signs are replaced as follows. Let the total number of degrees of freedom when the whole structure is divided into mesh and is expressed by the h-method element be n_(T), and eq. (943) is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 416} \right\rbrack & \; \\ {\underset{n_{T} \times 1}{\left\{ U \right\}} = {\underset{n_{T} \times 1}{\left\{ U_{p} \right\}} + {\underset{n_{T} \times n_{o}}{\left\lbrack \psi_{U} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}}} & (948) \end{matrix}$

Further, let the number of degrees of freedom of a certain element be n, and let a displacement be U_(A), which is used exclusively regarding the element, and then, eq. (166) is given as:

$\begin{matrix} {\underset{2 \times 1}{\left\{ u_{A} \right\}} \equiv {\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}\underset{n_{A} \times 1}{\left\{ U_{A} \right\}}}} & (949) \end{matrix}$

The displacement used exclusively regarding the element is extracted from eq. (948), and is written as:

$\begin{matrix} {\underset{n_{A} \times 1}{\left\{ U_{A} \right\}} = {\underset{n_{A} \times 1}{\left\{ U_{p\; A} \right\}} + {\underset{n_{A} \times n_{o}}{\left\lbrack \psi_{UA} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}}} & (950) \end{matrix}$

Substituting this into U_(A) of eq. (949) gives:

$\begin{matrix} {{\underset{2 \times 1}{\left\{ u_{A} \right\}} \equiv {\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}\underset{n_{A} \times 1}{\left\{ U_{A} \right\}}}} = {\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}\left( {\underset{n_{A} + 1}{\left\{ U_{p\; A} \right\}} + {\underset{n_{A} \times n_{o}}{\left\lbrack \psi_{UA} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}} \right)}} & (951) \end{matrix}$

We can see that this equation expresses the internal displacement distribution in the element, and constitutes a function of the mode coefficient {a_(o)}.

The simultaneous partial differential equation (23) of a certain element is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 417} \right\rbrack & \; \\ {{\sum\limits_{j}{L_{ij}u_{j}}} = {f_{i}({Aforementioned})}} & (23) \end{matrix}$

This is expressed with matrices gives as the following eq. (421):

$\begin{matrix} {{{\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times 1}{\left\{ u \right\}}} = \underset{2 \times 1}{\left\{ f \right\}}}({Aforementioned})} & (421) \end{matrix}$

Since the internal displacement of the element {u} of the h-method element is expressed with {u_(A)} of eq. (951), this equation is transformed to:

$\begin{matrix} {{\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}\left( {\underset{n_{A} + 1}{\left\{ U_{p\; A} \right\}} + {\underset{n_{A} \times n_{o}}{\left\lbrack \psi_{UA} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}} \right)} = \underset{2 \times 1}{\left\{ f \right\}}} & (952) \end{matrix}$

However there is no guarantee that the relationship indicated by the equal sign of this equation, and only what can be expected is that the relationship indicated by an approximately equal sign is established by giving an appropriate {a_(o)}. Therefore, the error function E_(k) with respect to the k-th element is defined as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 418} \right\rbrack & \; \\ {\underset{2 \times 1}{\left\{ E_{k} \right\}} \equiv {{\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}\left( {\underset{n_{A} + 1}{\left\{ U_{p\; A} \right\}} + {\underset{n_{A} \times n_{o}}{\left\lbrack \psi_{UA} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}} \right)} - \underset{2 \times 1}{\left\{ f \right\}}}} & (953) \end{matrix}$

Then, the differential equation (952) is transformed to:

$\begin{matrix} {\underset{2 \times 1}{\left\{ E_{k} \right\}} = \underset{2 \times 1}{\left\{ 0 \right\}}} & (954) \end{matrix}$

According to eq. (953), we can find how an internal error E_(k) in the element is distributed depending on the mode coefficient {a_(o)} The functional Π is given as the following eq. (128):

$\begin{matrix} {\Pi \equiv {\sum\limits_{i}{\int_{S}{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right)^{2}{{s({Aforementioned})}}}}}} & (128) \end{matrix}$

Therefore, given

$\begin{matrix} {{\Pi_{k} \equiv {\int_{S}{{\underset{1 \times 2}{\left\{ E_{k} \right\}}}^{T}\underset{2 \times 1}{\left\{ E_{k} \right\}}{s}}}},} & (955) \end{matrix}$

the mode coefficient {a_(o)} may be determined so that the variation thereof is zero, and this is equivalent to that the variational principle is applied. However, at this stage, the variation principle is applied to only one element. Let the total number of elements representing the whole structure be nE, and let the functional in the whole structure be Π_(τ). Then, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 419} \right\rbrack & \; \\ {{\prod\limits_{T}^{\;}{\equiv {\sum\limits_{k = 1}^{n_{E}}\; \prod\limits_{k}^{\;}}}} = {\sum\limits_{k = 1}^{n_{E}}\; {\int_{S}^{\;}{{\underset{1 \times 2}{\left\{ E_{k} \right\}}}^{T}\underset{2 \times 1}{\left\{ E_{k} \right\}}\ {s}}}}} & (956) \end{matrix}$

Here, the mode coefficient {a_(o)} is determined so that the variation of Π_(τ) is zero, and this is equivalent to that the variational principle is applied to the whole structure, and is equal to the equation (127) that expresses the direct variational method as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 420} \right\rbrack & \; \\ {{\sum\limits_{i}^{\;}\; {\int_{S}^{\;}{{\left( {{\sum\limits_{j}^{\;}\; {L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}{\sum\limits_{j}^{\;}\; {L_{ij}u_{j}\ {s}}}}}} = 0} & \underset{({Aforementioned})}{(127)} \end{matrix}$

In other words, the mode coefficient {a_(o)} may be determined so that the following is established:

$\begin{matrix} {{\frac{\delta \overset{\;}{\prod_{T}}}{2} \equiv {\sum\limits_{k = 1}^{n_{E}}\; {\int_{S}^{\;}{{\underset{1 \times 2}{\left\{ {\delta \; E_{k}} \right\}}}^{T}\underset{2 \times 1}{\left\{ E_{k} \right\}}\ {s}}}}} = 0} & (957) \end{matrix}$

Here, the following is established:

$\begin{matrix} {\underset{2 \times 1}{\left\{ {\delta \; E_{k}} \right\}} \equiv {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}\underset{n_{A} \times n_{o}}{\left\lbrack \psi_{UA} \right\rbrack}\underset{n_{o} \times 1}{\left\{ {\delta \; a_{o}} \right\}}}} & (958) \end{matrix}$

Therefore, the variation δΠ_(τ) is given as:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 421} \right\rbrack} & \; \\ {\frac{\delta \; \Pi_{T}}{2} \equiv {{\underset{1 \times n_{o}}{\left\{ {\delta \; a_{o}} \right\}^{T}}{\sum\limits_{k = 1}^{n_{E}}\; {\underset{n_{o} \times n_{A}}{\left\lbrack \psi_{UA} \right\rbrack^{T}}{\int_{S}^{\;}{\left( {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}} \right)^{T}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}\ {s}\underset{n_{A} \times n_{o}}{\left\lbrack \psi_{UA} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}}}}} + {\underset{1 \times n_{o}}{\left\{ {\delta \; a_{o}} \right\}^{T}}{\sum\limits_{k = 1}^{n_{E}}\; {\underset{n_{o} \times n_{A}}{\left\lbrack \psi_{UA} \right\rbrack^{T}}{\int_{S}^{\;}{\left( {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}} \right)^{T}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}\ {s}\underset{n_{A} \times 1}{\left\{ U_{pA} \right\}}}}}}} - {\underset{1 \times n_{o}}{\left\{ {\delta \; a_{o}} \right\}^{T}}{\sum\limits_{k = 1}^{n_{E}}\; {\underset{n_{o} \times n_{A}}{\left\lbrack \psi_{UA} \right\rbrack^{T}}{\int_{S}^{\;}{\left( {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}} \right)^{T}\ {s}\underset{2 \times 1}{\left\{ f \right\}}}}}}}}} & (959) \end{matrix}$

In order that the equation (957) should be established with respect to an arbitrary variation {δa_(o)}, when the following is given

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 422} \right\rbrack} & \; \\ {\mspace{79mu} {\underset{n_{o} \times n_{o}}{\lbrack A\rbrack} \equiv {\sum\limits_{k = 1}^{n_{E}}\; {\underset{n_{o} \times n_{A}}{\left\lbrack \psi_{UA} \right\rbrack^{T}}{\int_{S}^{\;}{\left( {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}} \right)^{T}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}{s}\underset{n_{A} \times n_{o}}{\left\lbrack \psi_{UA} \right\rbrack}}}}}}} & (960) \\ {\underset{n_{o} \times 1}{\left\{ B \right\}} \equiv {{\sum\limits_{k = 1}^{n_{E}}\; {\underset{n_{o} \times n_{A}}{\left\lbrack \psi_{UA} \right\rbrack^{T}}{\int_{S}^{\;}{\left( {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}} \right)^{T}{s}\underset{2 \times 1}{\left\{ f \right\}}}}}} - {\sum\limits_{k = 1}^{n_{E}}{\underset{n_{o} \times n_{A}}{\left\lbrack \psi_{UA} \right\rbrack^{T}}{\int_{S}^{\;}{\left( {\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}} \right)^{T}\underset{2 \times 2}{\lbrack L\rbrack}\underset{2 \times n_{A}}{\left\lbrack \chi_{A} \right\rbrack}\ {s}\underset{n_{A} \times 1}{\left\{ U_{pA} \right\}}}}}}}} & (961) \end{matrix}$

the following should be established.

$\begin{matrix} {{\underset{n_{o} \times n_{o}}{\lbrack A\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}} = \underset{n_{o} \times 1}{\left\{ B \right\}}} & (962) \end{matrix}$

This equation can be solved with respect to the mode coefficient {a_(o)}. We can consider that with this, a solution that satisfies the differential equations approximately as the whole structure system is obtained, and when the determined {a_(o)} is returned to eq. (938), the unknown displacement is settled, and when the same is returned to eq. (939), the unknown external force is settled. With this value taken as a starting point, a designer can proceed designing work, while confirming the state of feasible deformation, stress distribution, error distribution, and the like, by operating {a_(o)} with a slider.

11.3.12 Supplementary Description about Common Structural Characteristics

The simultaneous partial differential equation (23) is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 423} \right\rbrack & \; \\ {{\sum\limits_{j}^{\;}\; {L_{ij}u_{j}}} = f_{i}} & \underset{({Aforementioned})}{(23)} \end{matrix}$

In order to solve this, we use the primal simultaneous differential equation (105) and the dual simultaneous differential equation (106):

$\begin{matrix} {{\sum\limits_{j}^{\;}{L_{ij}\varphi_{j}}} = {\lambda \; w_{i}\varphi_{i}^{*}}} & \underset{({Aforementioned})}{(105)} \\ {{\sum\limits_{j}^{\;}{L_{ji}^{*}\varphi_{j}^{*}}} = {\lambda \; w_{i}\varphi_{i}}} & \underset{({Aforementioned})}{(106)} \end{matrix}$

Equations (105), (106) can be united into the following, with respect to a more general two-dimensional problem, though this can be applied to the static equilibrium equation of a two-dimensional elastic body:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 424} \right\rbrack & \; \\ {{\begin{bmatrix} 0 & 0 & L_{11}^{*} & L_{21}^{*} \\ 0 & 0 & L_{12}^{*} & L_{22}^{*} \\ L_{11} & L_{12} & 0 & 0 \\ L_{21} & L_{22} & 0 & 0 \end{bmatrix}\begin{Bmatrix} \varphi_{1} \\ \varphi_{2} \\ \varphi_{1}^{*} \\ \varphi_{2}^{*} \end{Bmatrix}} = {{\lambda \begin{bmatrix} w_{1} & 0 & 0 & 0 \\ 0 & w_{2} & 0 & 0 \\ 0 & 0 & w_{1} & 0 \\ 0 & 0 & 0 & w_{2} \end{bmatrix}}\begin{Bmatrix} \varphi_{1} \\ \varphi_{2} \\ \varphi_{1}^{*} \\ \varphi_{2}^{*} \end{Bmatrix}}} & (963) \end{matrix}$

In this equation, the matrix part of the left side of the equation is a self-adjoint operator. In other words, this equation is devised so that a non-self-adjoint problem is changed to a self-adjoint problem so that an eigenfunction of a complete system can be obtained. Since it is a two-dimensional problem, two equations exist in eq. (23), but unknown functions (also referred to as variables) are φ₁, φ₂, φ₁*, and φ₂*. Thus, four unknown functions, the number of which is twice the number of the equations, are defined. Further, as described in Section 8.18, analysis is performed using functions φ₁, φ₂, φ₁*, and φ₂* such that an eigenvalue is zero. Thus, this method can be used in the case of a unique solution and in the case of a plurality of solutions, and this is a structural characteristic.

On the other hand, in the formulation with respect to a conventional element, that is, the following eq. (442)

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 425} \right\rbrack & \; \\ {{{\begin{bmatrix} \underset{n \times n}{\lbrack K\rbrack} & \underset{n \times n}{\left\lbrack {- I} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n \times 1}{\left\{ U \right\}} \\ \underset{n \times 1}{\left\{ F \right\}} \end{Bmatrix}} = \underset{n \times 1}{\left\{ 0 \right\}}},} & \underset{({Aforementioned})}{(442)} \end{matrix}$

the number of equations is the number of rows “n”, and 2n variables U,F, the number of which is twice the number of the equations, are defined. Further, as described in Section 9.2, the underdetermined system is equivalent to a case where the eigenvalue λ is zero, that is, a solution is determined as the following eq. (455)

$\begin{matrix} {\underset{n_{v} \times 1}{\left\{ s_{v} \right\}} = {\underset{n_{v} \times 1}{\left\{ s_{p} \right\}} + {\underset{n_{v} \times n_{o}}{\left\lbrack \psi_{o} \right\rbrack}\underset{n_{o} \times 1}{\left\{ a_{o} \right\}}}}} & \underset{({Aforementioned})}{(455)} \end{matrix}$

Thus, this method can be used in the case of a unique solution and in the case of a plurality of solutions, and this is a structural characteristic.

These are characteristics shared by both of the solution methods.

11.4 Static Deflection of Ring

This section should better be included in Chapter 10.

11.4.1 Differential Equation

Static equilibrium equations of a two-dimensional elastic body are, in the orthogonal coordinate system, given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 426} \right\rbrack & \; \\ {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} u_{x}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}u_{y}}} = {{- \frac{1}{G}}b_{x}}} & \underset{({Aforementioned})}{(1)} \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}u_{x}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} u_{y}}} = {{- \frac{1}{G}}b_{y}}} & \underset{({Aforementioned})}{(2)} \end{matrix}$

where u_(x), u_(y) represent displacements in the x, y directions, respectively, b_(x), b_(y) represent body forces in the x, y directions per unit volume, and G represents a modulus of rigidity. Let a Poisson's ratio be v, and then, the constant μ is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 427} \right\rbrack & \; \\ {{\mu \equiv {\frac{1 + v}{1 - v}\left( {{Plane}\mspace{14mu} {Stress}} \right)}},{\mu \equiv {\frac{1}{1 - {2\; v}}\left( {{Plane}\mspace{14mu} {Strain}} \right)}},} & \underset{({Aforementioned})}{(3)} \end{matrix}$

Here, the right sides of differential equations (1), (2) are defined as:

$\begin{matrix} {\begin{Bmatrix} f_{x} \\ f_{y} \end{Bmatrix} \equiv {{- \frac{1}{G}}\begin{Bmatrix} b_{x} \\ b_{y} \end{Bmatrix}}} & (964) \end{matrix}$

We recognize this as a constant.

11.4.2 Coordinate Conversion

The relationship between the orthogonal coordinate system and the polar coordinate system is given as:

[Formula 428]

x=r cos θ  (965)

y=r sin θ  (966)

The inverse relationship of the same is as follows:

$\begin{matrix} {r = \sqrt{x^{2} + y^{2}}} & (967) \\ {\theta = {\arctan \frac{y}{x}}} & (968) \end{matrix}$

11.4.3 Displacement Conversion

The relationship among orthogonal coordinate components u_(x), u_(y) and polar coordinate components u_(r), u_(θ) of a displacement is given as:

[Formula 429]

u _(r)=cos θ·u _(x)+sin θ·u _(y)  (969)

u _(θ)=−sin θ·u _(x)+cos θ·u _(y)  (970)

The inverse relationship of the same is as follows:

u _(x)=cos θ·u _(r)+sin θ·u _(y)  (971)

u _(y)=sin θ·u _(r)+cos θ·u _(o)  (972)

11.4.4 Surface Force Conversion

Conversion of a surface force is similar to that of a displacement. The relationship among orthogonal coordinate components p_(x), p_(y) and polar coordinate components p_(r), p_(θ) of a surface force is given as:

[Formula 430]

p _(r)=cos θ·p _(x)+sin θ·p _(y)  (973)

p _(θ)=−sin θ·p _(x)+cos θ·p _(y)  (974)

The inverse relationship of the same is as follows:

p _(x)=cos θ·p _(r)−sin θ·p _(θ)  (975)

p _(y)=sin θ·p _(r)+cos θ·p _(θ)  (976)

11.4.5 Stress Conversion

The relationship among orthogonal coordinate components σ_(x), σ_(y), τ_(xy) and polar coordinate components σ_(r), σ_(θ), τ_(rθ) of stress is given as:

[Formula 431]

σ_(r)=cos² θ·σ_(x)+sin² θ·+2 sin θ cos θ·τ_(xy)  (977)

σ_(θ)=sin² θ·σ cos² θ·σ_(y)2 sin θ cos θ·τ_(xy)  (978)

τ_(rθ)=sin θ cos θ·(σ_(y)−σ_(x))+(cos² θ−sin² θ)·τ_(xy)  (979)

The inverse relationship of the same is as follows:

σ_(x)=cos² θ·σ_(r)+sin² θ·σ_(θ)−2 sin θ cos θ·τ_(rθ)  (980)

[Formula 432]

σ_(y)=sin² θ·σ_(r)+cos² θ·σ_(θ)+2 sin θ cos θ·τ_(rθ)  (981)

τ_(xy)=sin θ cos θ·(σ_(r)−σ_(θ))+(cos² θ−sin² θ)·τ_(rθ)  (982)

11.4.6 Conversion of Differential Operator

Differentiating eq. (967) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 433} \right\rbrack & \; \\ {{\frac{\partial r}{\partial x} \equiv \frac{x}{\sqrt{x^{2} + y^{2}}}} = {\frac{x}{r} = {\cos \; \theta}}} & (983) \\ {{\frac{\partial r}{\partial y} \equiv \frac{y}{\sqrt{x^{2} + y^{2}}}} = {\frac{y}{r} = {\sin \; \theta}}} & (984) \end{matrix}$

Differentiating eq. (968) gives:

$\begin{matrix} {{\frac{\partial\theta}{\partial x} \equiv {- \frac{y}{x^{2} + y^{2}}}} = {{- \frac{y}{r^{2}}} = {{- \frac{1}{r}}\sin \; \theta}}} & (985) \\ {{\frac{\partial\theta}{\partial y} \equiv {+ \frac{x}{x^{2} + y^{2}}}} = {{+ \frac{x}{r^{2}}} = {{+ \frac{1}{r}}\cos \; \theta}}} & (986) \end{matrix}$

According to the chain rule, the following is established with respect to an arbitrary function φ:

$\begin{matrix} {\frac{\partial\varphi}{\partial x} \equiv {{\frac{\partial r}{\partial x}\frac{\partial\varphi}{\partial r}} + {\frac{\partial\theta}{\partial x}\frac{\partial\varphi}{\partial\theta}}} \equiv {{\cos \; \theta \frac{\partial\varphi}{\partial r}} - {\frac{1}{r}\sin \; \theta \frac{\partial\varphi}{\partial\theta}}}} & (987) \\ {\frac{\partial\varphi}{\partial y} \equiv {{\frac{\partial r}{\partial y}\frac{\partial\varphi}{\partial r}} + {\frac{\partial\theta}{\partial y}\frac{\partial\varphi}{\partial\theta}}} \equiv {{\sin \; \theta \frac{\partial\varphi}{\partial r}} - {\frac{1}{r}\cos \; \theta \frac{\partial\varphi}{\partial\theta}}}} & (988) \end{matrix}$

Differentiating these again gives:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 434} \right\rbrack} & \; \\ {\frac{\partial^{2}\varphi}{\partial x^{2}} \equiv {{\cos^{2}{\theta \cdot \frac{\partial^{2}\varphi}{\partial r^{2}}}} + {\frac{1}{r}\sin^{2}{\theta \cdot \frac{\partial\varphi}{\partial r}}} + {\frac{1}{r^{2}}\sin^{2}{\theta \cdot \frac{\partial^{2}\varphi}{\partial\theta^{2}}}} - {\sin \; 2\; {\theta \cdot {\frac{\partial}{\partial r}\left\lbrack {\frac{1}{r}\frac{\partial\varphi}{\partial\theta}} \right\rbrack}}}}} & (989) \\ {\frac{\partial^{2}\varphi}{\partial y^{2}} \equiv {{\sin^{2}{\theta \cdot \frac{\partial^{2}\varphi}{\partial r^{2}}}} + {\frac{1}{r}\cos^{2}{\theta \cdot \frac{\partial\varphi}{\partial r}}} + {\frac{1}{r^{2}}\cos^{2}{\theta \cdot \frac{\partial^{2}\varphi}{\partial\theta^{2}}}} + {\sin \; 2\; {\theta \cdot {\frac{\partial}{\partial r}\left\lbrack {\frac{1}{r}\frac{\partial\varphi}{\partial\theta}} \right\rbrack}}}}} & (990) \\ {\frac{\partial^{2}\varphi}{{\partial x}{\partial y}} \equiv {{\sin \; \theta \; \cos \; {\theta \cdot \left( {\frac{\partial^{2}\varphi}{\partial r^{2}} - {\frac{1}{r}\frac{\partial\varphi}{\partial r}} - {\frac{1}{r^{2}}\frac{\partial^{2}\varphi}{\partial\theta^{2}}}} \right)}} + {\cos \; 2\; {\theta \cdot {\frac{\partial}{\partial r}\left\lbrack {\frac{1}{r}\frac{\partial\varphi}{\partial\theta}} \right\rbrack}}}}} & (991) \end{matrix}$

Adding eqs. (989) and (990) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 435} \right\rbrack & \; \\ {{{\nabla^{2}\varphi} \equiv {\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)\varphi}} = {\left( {\frac{\partial^{2}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}}} \right)\varphi}} & (992) \end{matrix}$

This represents a change of variables of the Laplace operator (Laplacian) ∇².

11.4.7 Biharmonic Equation

The following biharmonic equations are obtained from the simultaneous differential equations (1) and (2):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 436} \right\rbrack & \; \\ {{{\nabla^{4}u_{x}} \equiv {\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)^{2}u_{x}}} = 0} & (993) \\ {{{\nabla^{4}u_{y}} \equiv {\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)^{2}u_{y}}} = 0} & (994) \end{matrix}$

Expanding operators gives:

$\begin{matrix} {{\nabla^{4}{\equiv \left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)^{2}}} = {\frac{\partial^{4}}{\partial x^{4}} + {2\frac{\partial^{4}}{{\partial x^{2}}{\partial y^{2}}}} + \frac{\partial^{4}}{\partial y^{4}}}} & (995) \end{matrix}$

According to eq. (992), the differential equations (993) and (994) are converted to the following in the polar coordinate system:

$\begin{matrix} {{{\nabla^{4}u_{x}} \equiv {\left( {\frac{\partial^{2}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}}} \right)^{2}u_{x}}} = 0} & (996) \\ {{{\nabla^{4}u_{y}} \equiv {\left( {\frac{\partial^{2}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}}} \right)^{2}u_{y}}} = 0} & (997) \end{matrix}$

Expanding operators gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 437} \right\rbrack & \; \\ \begin{matrix} {\nabla^{4}{\equiv \left( {\frac{\partial^{2}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}}} \right)^{2}}} \\ {= {\left( {{\frac{1}{r^{3}}\frac{\partial}{\partial r}} - {\frac{1}{r^{2}}\frac{\partial^{2}}{\partial r^{2}}} + {\frac{2}{r}\frac{\partial^{3}}{\partial r^{3}}} + \frac{\partial^{4}}{\partial r^{4}}} \right) +}} \\ {{{\frac{2}{r^{2}}\left( {{{- \frac{1}{r}}\frac{\partial}{\partial r}} + \frac{\partial^{2}}{\partial r^{2}}} \right)\frac{\partial^{2}}{\partial\theta^{2}}} + {\frac{1}{r^{4}}\left( {{4\frac{\partial^{2}}{\partial\theta^{2}}} + \frac{\partial^{4}}{\partial\theta^{4}}} \right)}}} \end{matrix} & (998) \end{matrix}$

Let general solutions of differential equations (996) and (997) of the polar coordinate system be ψ(r,θ), a solution satisfying eq. (999) is given as eq. (1000) in the variable separation form:

[Formula 438]

∇⁴ψ=0  (999)

ψ(r,θ)≡A(r)·B(θ)  (1000)

11.4.8 Solution by Separation of Variables

Substituting eq. (1000) into eq. (999) gives:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 439} \right\rbrack} & \; \\ {{{{\nabla^{4}\psi} \equiv {\frac{1}{r^{4}}\left\lbrack {{2\; {r\left( {{- A^{\&}} + {r\text{?}}} \right)}\text{?}} + {A\left( {{4\text{?}} + \text{?}} \right)} + {{rB}\left\{ {A^{\&} + {r\left( {{- \text{?}} + {2\; r\text{?}} + {r^{2}\text{?}}} \right)}} \right\}}} \right\rbrack}} = 0}{\text{?}\text{indicates text missing or illegible when filed}}} & (1001) \end{matrix}$

With periodicity of 2,r being expected, B(6) is given as:

B(θ)≡sin(mθ),cos(mθ)  (1002)

Here, m represents an integer. Therefore, we obtain:

$\begin{matrix} {{\text{?} = {{- m^{2}}B}}\mspace{79mu} {\text{?} = {{{- m^{2}}\text{?}} = {m^{4}B}}}{\text{?}\text{indicates text missing or illegible when filed}}} & (1003) \end{matrix}$

Substituting these into eq. (1001) and rearranging the same, we obtain:

$\begin{matrix} {{{{{m^{2}\left( {m^{2} - 4} \right)}A} + {r\left\lbrack {{\left( {1 + {2m^{2}}} \right)A\text{?}} + {r\left\{ {{{- \left( {1 + {2m^{2}}} \right)}\text{?}} + {r\left( {{2\text{?}} + {r\text{?}}} \right)}} \right\}}} \right\rbrack}} = 0}{\text{?}\text{indicates text missing or illegible when filed}}} & (1004) \end{matrix}$

A general solution of this equation is given as:

[Formula 440]

A(r)=r ^(2−m) ,r ^(−m) ,r ^(m) ,r ^(2+m)  (1005)

As a result, a product of eq. (1002) and eq. (1005) as follows is a solution by separation of variables.

$\begin{matrix} {{\psi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} r^{2 - m} \\ r^{- m} \\ r^{m} \\ r^{2 + m} \end{pmatrix}}} & (1006) \end{matrix}$

Incidentally, as inconveniences occur in the case where m=0 in the foregoing equation, the following is given:

[Formula 441]

B(θ)≡1,θ  (1007)

However, as B(θ)≡θ is inconvenient from the viewpoint of periodicity, this is excluded, and with only B(θ)≡1 as an object of solution, we obtain:

$\begin{matrix} {\mspace{79mu} {{\text{?} = 0}\mspace{79mu} {\text{?} = 0}{\text{?}\text{indicates text missing or illegible when filed}}}} & (1008) \end{matrix}$

Substituting these into eq. (1001) and rearranging the same, we obtain:

$\begin{matrix} {\mspace{79mu} {{{A^{\&} + {r\left\{ {{- \text{?}} + {r\left( {{2\text{?}} + {r\text{?}}} \right)}} \right\}}} = 0}{\text{?}\text{indicates text missing or illegible when filed}}}} & (1009) \end{matrix}$

The general solution of this equation is:

A(r)=1,r ²,ln r,r ² ln r  (1010)

As a result, we obtain the following as a solution by separation of variables.

$\begin{matrix} {{\psi \left( {r,\theta} \right)} \equiv {1 \times \begin{pmatrix} 1 \\ r^{2} \\ {\ln \; r} \\ {r^{2}\ln \; r} \end{pmatrix}}} & (1011) \end{matrix}$

Further, since inconveniences occur also in the case where m=1 in eq. (1006), the following is given:

[Formula 442]

B(θ)≡sin(θ),cos(θ)  (1012)

Then, we obtain:

$\begin{matrix} {\mspace{79mu} {{\text{?} = {- B}}\mspace{79mu} {\text{?} = {{- \text{?}} = B}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (1013) \end{matrix}$

Substituting these into (1001) and rearranging the same, we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 443} \right\rbrack} & \; \\ {\mspace{79mu} {{{{{- 3}\; A} + {r\left\lbrack {{3A^{\&}} + {r\left\{ {{{- 3}\text{?}} + {r\left( {{2\text{?}} + {r\text{?}}} \right)}} \right\}}} \right\rbrack}} = 0}{\text{?}\text{indicates text missing or illegible when filed}}}} & (1014) \end{matrix}$

The general solution of this equation is:

$\begin{matrix} {{{A(r)} = \frac{1}{r}},r,r^{3},{r\; \ln \; r}} & (1015) \end{matrix}$

As a result, a product of eq. (1012) and eq. (1015) as follows is a solution by separation of variables.

$\begin{matrix} {{\psi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin (\theta)} \\ {\cos (\theta)} \end{pmatrix} \times \begin{pmatrix} \frac{1}{r} \\ r \\ r^{3} \\ {r\; \ln \; r} \end{pmatrix}}} & \left( 1016 \right. \end{matrix}$

In view of the foregoing description, eqs. (1006), (1011), and (1016) may be used as the format of the solution of the differential equation (1001).

11.4.9 Boundary Condition

Let an outer radius and an inner radius of a two-dimensional ring be R and γR, respectively. The range of an inner diameter ratio γ is as follows:

[Formula 444]

0<γ<1  (1017)

How to support an outer edge in a ring on which uniform gravity is acting, in order to provide a boundary condition of an inner edge such that the surface force is zero and the displacement is zero is determined. This is a problem that has been considered insoluble conventionally.

A boundary condition such that a displacement of an inner edge is zero is given as:

[Formula 445]

u _(r)|_(r=γR)=0

u _(θ)|_(r=γR)=0  (1018)

A boundary condition such that a surface force of an inner edge is zero is given as:

σ_(r)|_(r=γR)=0

τ_(rθ)|_(r=γR)=0  (1019)

11.4.10 Analytical Solution

In the case of a solution that satisfies the differential equations (1), (2) and the boundary condition of eqs. (1018) and (1019), the format of eq. (1016) is unnecessary. Therefore, general solutions u_(x) and u_(y) of the differential equations (996) and (997) are expressed by eqs. (1006) and (1011), respectively. As a result of calculation, the functions are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 446} \right\rbrack & \; \\ {\chi_{1} \equiv {\frac{2 + \mu}{8\left( {1 + \mu} \right)}\left\{ {{\left( {\frac{r}{R} - \gamma} \right)\left( {\frac{r}{R} + \gamma} \right)} - {2{\gamma^{2}\left( {{\ln \; \frac{r}{R}} - {\ln \; \gamma}} \right)}}} \right\}}} & (1020) \\ {\chi_{2} \equiv {\frac{\mu}{16\left( {1 + \mu} \right)}\left( {\frac{r}{R} - {\gamma^{2}\frac{R}{r}}} \right)^{2}\cos \; 2\theta}} & (1021) \\ {\chi_{3} \equiv {\frac{\mu}{16\left( {1 + \mu} \right)}\left( {\frac{r}{R} - {\gamma^{2}\frac{R}{r}}} \right)^{2}\sin \; 2\theta}} & (1022) \end{matrix}$

Then, we obtain:

u _(x) ≡R ² {f _(x)(χ₁−χ₂)−f _(y)χ₃}  (1023)

u _(y) ≡R ² {−f _(x)χ₃ +f _(y)(χ₁+χ₂)}  (1024)

Here, f_(x),f_(y) are given as the following eq. (964):

$\begin{matrix} {{\begin{Bmatrix} f_{x} \\ f_{y} \end{Bmatrix} \equiv {{- \frac{1}{G}}\begin{Bmatrix} b_{x} \\ b_{y\;} \end{Bmatrix}}}({Aforementioned})} & (964) \end{matrix}$

where b_(x), b_(y) represent body forces in the x, y directions per unit volume, and G represents a modulus of rigidity. As f_(x), f_(y) have a dimensionality of the reciprocal number of the length, the outer radius R is assumed to be the characteristic length, and dimensionless load coefficients c_(x), c_(y) are defined as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 447} \right\rbrack & \; \\ \left. {{\frac{1}{R}\begin{Bmatrix} c_{x} \\ c_{y} \end{Bmatrix}} \equiv \begin{Bmatrix} f_{x} \\ f_{y\;} \end{Bmatrix} \equiv {{- \frac{1}{G}}\begin{Bmatrix} b_{x} \\ b_{y} \end{Bmatrix}}}\Rightarrow{\begin{Bmatrix} c_{x} \\ c_{y} \end{Bmatrix} \equiv {{- \frac{R}{G}}\begin{Bmatrix} b_{x} \\ b_{y} \end{Bmatrix}}} \right. & (1025) \end{matrix}$

This facilitates the handling of the problem.

In a plane stress state, the material constant is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 448} \right\rbrack & \; \\ {{v \equiv \frac{3}{10}},{{\mu \equiv \frac{1 + v}{1 - v}} = \frac{13}{7}}} & (1026) \end{matrix}$

The inner diameter ratio γ is given as:

$\begin{matrix} {\gamma \equiv \frac{3}{10}} & (1027) \end{matrix}$

Deformation and stress distribution in the case where a uniform gravity load f_(y) is caused to act thereon is shown in FIG. 42. Here, the load coefficients c_(x), c_(y) are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 449} \right) & \; \\ {\begin{Bmatrix} c_{x} \\ c_{y} \end{Bmatrix} \equiv \begin{Bmatrix} 0 \\ 1 \end{Bmatrix}} & (1028) \end{matrix}$

Thus, the displacement is made dimensionless as to the outer radius R.

11.4.11 Formula of Partial Integration

As described in Section 11.2.1, partial integration using arbitrary functions f, f* as functions of x, y gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 450} \right\rbrack & \; \\ {{{\int_{S}{\frac{\partial f}{\partial x}f^{*}{s}}} = {{\int_{C}{n_{x}{f \cdot f^{*}}{c}}} - {\int_{S}{f\frac{\partial f^{*}}{\partial x}{s}}}}}{and}} & (816) \\ {{\int_{S}{\frac{\partial f}{\partial y}f^{*}{s}}} = {{\int_{C}{n_{y}{f \cdot f^{*}}{c}}} - {\int_{S}{f\frac{\partial f^{*}}{\partial y}{{s({Aforementioned})}}}}}} & (817) \end{matrix}$

Expanding these gives:

$\begin{matrix} {{{\int_{S}{\frac{\partial^{2}f}{\partial x^{2}}f^{*}{s}}} = {{\int_{C}{\left( {{n_{x}\frac{\partial f}{\partial x}f^{*}} - {n_{s}f\frac{\partial f^{*}}{\partial x}}} \right){c}}} + {\int_{S}{f\frac{\partial^{2}f^{*}}{\partial x^{2}}{s}}}}}\mspace{20mu} {and}} & (1029) \\ {{\int_{S}{\frac{\partial^{2}f}{\partial y^{2}}f^{*}{s}}} = {{\int_{C}{\left( {{n_{y}\frac{\partial f}{\partial y}f^{*}} - {n_{y}f\frac{\partial f^{*}}{\partial y}}} \right){c}}} + {\int_{S}{f\frac{\partial^{2}f^{*}}{\partial y^{2}}{s}}}}} & (1030) \end{matrix}$

Further, this also gives:

$\begin{matrix} {\mspace{20mu} \left\lbrack {{Formula}\mspace{14mu} 451} \right\rbrack} & \; \\ {{{\int_{S}{\frac{\partial^{2}f}{{\partial x}{\partial y}}f^{*}{s}}} = {{\int_{C}{\left( {{n_{x}\frac{\partial f}{\partial y}f^{*}} - {n_{y}f\frac{\partial f^{*}}{\partial x}}} \right){c}}} + {\int_{S}{f\frac{\partial^{2}f^{*}}{{\partial x}{\partial y}}{s}}}}}\mspace{20mu} {and}} & (1031) \\ {{\int_{S}{\frac{\partial^{2}f}{{\partial x}{\partial y}}f^{*}{s}}} = {{\int_{C}{\left( {{n_{y}\frac{\partial f}{\partial x}f^{*}} - {n_{x}\frac{\partial f^{*}}{\partial y}}} \right){c}}} + {\int_{S}{f\frac{\partial^{2}f^{*}}{{\partial x}{\partial y}}{s}}}}} & (1032) \end{matrix}$

Equating eq. (1031) and eq. (1032) gives:

$\begin{matrix} {{\int_{C}{\left( {{n_{x}\frac{\partial f}{\partial y}f^{*}} - {n_{y}f\frac{\partial f^{*}}{\partial x}}} \right){c}}} = {\int_{C}{\left( {{n_{y}\frac{\partial f}{\partial x}f^{*}} - {n_{x}f\frac{\partial f^{*}}{\partial y}}} \right)\; {c}}}} & (1033) \end{matrix}$

Rearranging these, we obtain the relationship of boundary integration as follows:

$\begin{matrix} {{\int_{C}{\left( {{n_{x}\frac{\partial f}{\partial y}} - {n_{y}\frac{\partial f}{\partial x}}} \right)f^{*}{c}}} = {\int_{C}{{f\left( {{n_{y\;}\frac{\partial f^{*}}{\partial x}} - {n_{x}\frac{\partial f^{*}}{\partial y}}} \right)}{c}}}} & (1034) \end{matrix}$

11.4.12 Adjoint Differential Operator and Adjoint Boundary Condition

Expressing the right sides of the differential equations (1) and (2) with eq. (964) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 452} \right\rbrack & \; \\ {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2\;}}} \right\} u_{x}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}u_{y}}} = f_{x}} & (1035) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}u_{x}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} u_{y}}} = f_{y}} & (1036) \end{matrix}$

General description of these gives eq. (23) as follows:

$\begin{matrix} {{\sum\limits_{j}{L_{ij}u_{j}}} = {f_{i}({Aforementioned})}} & (23) \end{matrix}$

Here, the original differential operator L_(ij) is expressed by eq. (15) as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 453} \right\rbrack & \; \\ {{\begin{bmatrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{bmatrix} \equiv \begin{bmatrix} {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} & {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \\ {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} & {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \end{bmatrix}}({Aforementioned})} & (15) \end{matrix}$

The adjoint differential operator L_(ij)* is expressed by eq. (16) as follows:

$\begin{matrix} {{\begin{bmatrix} L_{11}^{*} & L_{21}^{*} \\ L_{12}^{*} & L_{22}^{*} \end{bmatrix} \equiv \begin{bmatrix} {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} & {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \\ {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} & {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \end{bmatrix}}({Aforementioned})} & (16) \end{matrix}$

A sum of integration obtained by multiplying eq. (23) by the dual displacement u_(i)*, that is, inner product, is expressed by eq. (58) as follows:

$\begin{matrix} {{\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{L_{ij}{u_{j} \cdot u_{i}^{*}}\ {s}}}}} = {\sum\limits_{i}\; {\int_{S}{{f_{i} \cdot u_{i}^{*}}\ {s}}}}} & \underset{({Aforementioned})}{\mspace{146mu} (58)} \end{matrix}$

As preparation for the dual displacement u_(i)*, the displacement-strain relationships corresponding to eqs. (4), (5), and (6) are as follows, both regarding the plane stress and regarding the plane strain state:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 454} \right\rbrack & \; \\ {ɛ_{x}^{*} \equiv {\frac{\partial}{\partial x}u_{x}^{*}}} & (1037) \\ {ɛ_{y}^{*} \equiv {\frac{\partial}{\partial y}u_{y}^{*}}} & (1038) \\ {\gamma_{xy}^{*} \equiv {{\frac{\partial}{\partial y}u_{x}^{*}} + {\frac{\partial}{\partial x}u_{y}^{*}}}} & (1039) \end{matrix}$

The stress-strain relationships corresponding to eqs. (7), (8), (9) are as follows:

σ_(x) *=G{(μ+1)ε_(x)*+(μ−1)ε_(y)*}  (1040)

σ_(y) *=G{(μ−1)ε_(x)*+(μ+1)ε_(y)*}  (1041)

τ_(xy) *=Gγ _(xy)*  (1042)

Stress components corresponding to eqs. (10), (11), (12) are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 455} \right\rbrack & \; \\ {\sigma_{x}^{*} \equiv {G\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{1}^{*}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial u_{2}^{*}}{\partial y}}} \right\}}} & (1043) \\ {\sigma_{y}^{*} \equiv {G\left\{ {{\left( {\mu - 1} \right)\frac{\partial u_{1}^{*}}{\partial x}} + {\left( {\mu + 1} \right)\frac{\partial u_{2}^{*}}{\partial y}}} \right\}}} & (1044) \\ {\tau_{xy}^{*} = {\tau_{yx}^{*} \equiv {G\left( {\frac{\partial u_{2}^{*}}{\partial x} + \frac{\partial u_{1}^{*}}{\partial y}} \right)}}} & (1045) \end{matrix}$

Surface forces corresponding to eqs. (13), (14) are given as:

p _(x) *≡n _(x)σ_(x) *+n _(y)τ_(yx)*  (1046)

p _(y) *≡n _(x)τ_(xy) *+n _(y)σ_(y)*  (1047)

Here, partial integration of the left side of eq. (58) is performed.

-   -   When |i=1, j=1|, using eqs. (1029), (1030), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 456} \right\rbrack & \; \\ \begin{matrix} {{\int_{S}{L_{11}{u_{1} \cdot u_{1}^{*}}\ {s}}} \equiv {\int_{S}{\left\lbrack {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\rbrack {u_{1} \cdot u_{1}^{*}}\ {s}}}} \\ {= {{\left( {1 + \mu} \right){\int_{S}{{\frac{\partial^{2}u_{1}}{\partial x^{2}} \cdot u_{1}^{*}}\ {s}}}} + {\int_{S}{{\frac{\partial^{2}u_{1}}{\partial y^{2}} \cdot u_{1}^{*}}\ {s}}}}} \\ {= {{\int_{C}{\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{1}}{\partial x}} + {n_{y}\frac{\partial u_{1}}{\partial y}}} \right\} u_{1}^{*}\ {c}}} -}} \\ {{{\int_{C}{u_{1}\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{1}^{*}}{\partial x}} + {n_{y}\frac{\partial u_{1}^{*}}{\partial y}}} \right\} {c}}} +}} \\ {{\int_{S}{{u_{1} \cdot \left\lbrack {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\rbrack}u_{1}^{*}{s}}}} \end{matrix} & (1048) \end{matrix}$

Let the boundary term of the right side of this equation be R₁₁, let the differential operator of the same be L₁₁*, and they are given as:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 457} \right\rbrack} & \; \\ {{R_{11} \equiv {{\int_{C}{\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{1}}{\partial x}} + {n_{y}\frac{\partial u_{1}}{\partial y}}} \right\} u_{1}^{*}\ {c}}} - {\int_{C}{u_{1}\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{1}^{*}}{\partial x}} + {n_{y}\frac{\partial u_{1}^{*}}{\partial y}}} \right\} {c}}}}}\mspace{20mu} {L_{11}^{*} \equiv {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}}}} & (1049) \end{matrix}$

L₁₁* represents an adjoint differential operator shown in eq. (16).

-   -   When |i=1, j=2|, using eq. (1031), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 458} \right\rbrack & \; \\ \begin{matrix} {{\int_{S}{L_{12}{u_{2} \cdot u_{1}^{*}}\ {s}}} \equiv {\int_{S}{\left\lbrack {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \right\rbrack {u_{2} \cdot u_{1}^{*}}\ {s}}}} \\ {= {{\int_{C}{{\mu \left( {{n_{x}\frac{\partial u_{2}}{\partial y}u_{1}^{*}} - {n_{y}u_{2}\frac{\partial u_{1}^{*}}{\partial x}}} \right)}\ {c}}} +}} \\ {{\int_{S}{{u_{2} \cdot \left\lbrack {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \right\rbrack}u_{1}^{*}{s}}}} \end{matrix} & (1050) \end{matrix}$

Let the boundary term of the right side of this equation be R₁₂, let the differential operator of the same be L₁₂* and they are given as:

$\begin{matrix} {{R_{12} \equiv {\int_{C}{{\mu \left( {{n_{x}\frac{\partial u_{2}}{\partial y}u_{1}^{*}} - {n_{y}u_{2}\frac{\partial u_{1}^{*}}{\partial x}}} \right)}\ {c}}}}{L_{12}^{*} \equiv {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}}}} & (1051) \end{matrix}$

L₁₂* is an adjoint differential operator shown in eq. (16).

-   -   When |i=2, j=1|, using eq. (1032), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 459} \right\rbrack & \; \\ \begin{matrix} {{\int_{S}{L_{21}{u_{1} \cdot u_{2}^{*}}\ {s}}} \equiv {\int_{S}{\left\lbrack {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \right\rbrack {u_{1} \cdot u_{2}^{*}}\ {s}}}} \\ {= {{\int_{C}{{\mu \left( {{n_{y}\frac{\partial u_{1}}{\partial x}u_{2}^{*}} - {n_{x}u_{1}\frac{\partial u_{2}^{*}}{\partial y}}} \right)}\ {c}}} +}} \\ {{\int_{S}{{u_{1} \cdot \left\lbrack {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \right\rbrack}u_{2}^{*}{s}}}} \end{matrix} & (1052) \end{matrix}$

Let the boundary term of the right side of this equation be R₂₁, let the differential operator of the same be L₂₁, and they are given as:

$\begin{matrix} {{R_{21} \equiv {\int_{C}{{\mu \left( {{n_{y}\frac{\partial u_{1}}{\partial x}u_{2}^{*}} - {n_{x}u_{1}\frac{\partial u_{2}^{*}}{\partial y}}} \right)}\ {c}}}}{L_{21}^{*} \equiv {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}}}} & (1053) \end{matrix}$

L₂₁* represents an adjoint differential operator shown in eq. (16).

-   -   When |i=2, j=2|, using eqs. (1029), (1030), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 460} \right\rbrack & \; \\ \begin{matrix} {{\int_{S}{L_{22}{u_{2} \cdot u_{2}^{*}}\ {s}}} \equiv {\int_{S}{\left\lbrack {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\rbrack {u_{2} \cdot u_{2}^{*}}\ {s}}}} \\ {= {{\int_{S}{{\frac{\partial^{2}u_{2}}{\partial x^{2}} \cdot u_{2}^{*}}\ {s}}} + {\left( {1 + \mu} \right){\int_{S}{{\frac{\partial^{2}u_{2}}{\partial y^{2}} \cdot u_{2}^{*}}\ {s}}}}}} \\ {= {{\int_{C}{\left\{ {{n_{x}\frac{\partial u_{2}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{2}}{\partial y}}} \right\} u_{2}^{*}\ {c}}} -}} \\ {{{\int_{C}{u_{2}\left\{ {{n_{x}\frac{\partial u_{2}^{*}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{2}^{*}}{\partial y}}} \right\} {c}}} +}} \\ {{\int_{S}{{u_{2} \cdot \left\lbrack {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\rbrack}u_{2}^{*}{s}}}} \end{matrix} & (1054) \end{matrix}$

Let the boundary term of the right side of this equation be R₂₂, let the differential operator of the same be L₂₂*, and they are given as:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 461} \right\rbrack} & \; \\ {{R_{22} \equiv {{\int_{C}{\left\{ {{n_{x}\frac{\partial u_{2}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{2}}{\partial y}}} \right\} u_{2}^{*}\ {c}}} - {\int_{C}{u_{2}\left\{ {{n_{x}\frac{\partial u_{2}^{*}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{2}^{*}}{\partial y}}} \right\} {c}}}}}\mspace{20mu} {L_{22}^{*} \equiv {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}}}} & (1055) \end{matrix}$

L₂₂ is an adjoint differential operator given in eq. (16).

In the case where the following is satisfied as in the present case, the operators are referred to as self-adjoint differential operators:

L _(ji) *=L _(ij)  (17) (Aforementioned)

Adding the boundary terms and expressing the same as R, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 462} \right\rbrack & \; \\ \begin{matrix} {R \equiv {R_{11} + R_{12} + R_{21} + R_{22}}} \\ {= {{\int_{C}{\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{1}}{\partial x}} + {n_{y}\frac{\partial u_{1}}{\partial y}} + {\mu \; n_{x}\frac{\partial u_{2}}{\partial y}}} \right\} u_{1}^{*}\ {c}}} +}} \\ {{{\int_{C}{\left\{ {{n_{x}\frac{\partial u_{2}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{2}}{\partial y}} + {\mu \; n_{y}\frac{\partial u_{1}}{\partial x}}} \right\} u_{2}^{*}\ {c}}} -}} \\ {{{\int_{C}{u_{1}\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{1}^{*}}{\partial x}} + {n_{y}\frac{\partial u_{1}^{*}}{\partial y}} + {\mu \; n_{x}\frac{\partial u_{2}}{\partial x}}} \right\} {c}}} -}} \\ {{\int_{C}{u_{2}\left\{ {{n_{x}\frac{\partial u_{2}^{*}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{2}^{*}}{\partial y}} + {\mu \; n_{x}\frac{\partial u_{1}}{\partial x}}} \right\} {c}}}} \end{matrix} & (1056) \end{matrix}$

Transforming this equation and utilizing the relationship of boundary integration expressed by eq. (1034), we obtain:

$\begin{matrix} {R \equiv {{\int_{C}{\left\lbrack {{n_{x}\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{1}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial u_{2}}{\partial y}}} \right\}} + {n_{y}\left( {\frac{\partial u_{2}}{\partial x} + \frac{\partial u_{1}}{\partial y}} \right)}} \right\rbrack u_{1}^{*}\ {c}}} + {\int_{C}{\left\lbrack {{n_{x}\left( {\frac{\partial u_{2}}{\partial x} + \frac{\partial u_{1}}{\partial y}} \right)} + {n_{y}\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{2}}{\partial y}} + {\left( {\mu - 1} \right)\frac{\partial u_{1}}{\partial x}}} \right\}}} \right\rbrack u_{2}^{*}\ {c}}} - {\int_{C}{{u_{1}\left\lbrack {{n_{x}\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{1}^{*}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial u_{2}^{*}}{\partial y}}} \right\}} + {n_{y}\left( {\frac{\partial u_{2}^{*}}{\partial x} + \frac{\partial u_{1}^{*}}{\partial y}} \right)}} \right\rbrack}\ {c}}} - {\int_{C}{{u_{2}\left\lbrack {{n_{x}\left( {\frac{\partial u_{2}^{*}}{\partial x} + \frac{\partial u_{1}^{*}}{\partial y}} \right)} + {n_{y}\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{2}^{*}}{\partial y}} + {\left( {\mu - 1} \right)\frac{\partial u_{1}^{*}}{\partial x}}} \right\}}} \right\rbrack}\ {c}}}}} & (1057) \end{matrix}$

Using eqs. (10), (11), (12) and eqs. (1043), (1044), (1045), we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 463} \right\rbrack} & \; \\ {R \equiv {{\frac{1}{G}{\int_{C}{\left( {{n_{x}\sigma_{x}} + {n_{y}\tau_{yx}}} \right)u_{1}^{*}\ {c}}}} + {\frac{1}{G}{\int_{C}{\left( {{n_{x}\tau_{xy}} + {n_{y}\sigma_{y}}} \right)u_{2}^{*}\ {c}}}} - {\frac{1}{G}{\int_{C}{{u_{1}\left( {{n_{x}\sigma_{x}^{*}} + {n_{y}\tau_{yx}^{*}}} \right)}\ {c}}}} - {\frac{1}{G}{\int_{C}{{u_{2}\left( {{n_{x}\tau_{xy}^{*}} + {n_{y}\sigma_{y}^{*}}} \right)}\ {c}}}}}} & (1058) \end{matrix}$

Using eqs. (13), (14) and eqs. (1046), (1047), that is, Cauchy's formula, we obtain:

$\begin{matrix} {R \equiv {{\frac{1}{G}{\int_{C}{\left( {{p_{x}u_{1}^{*}} + {p_{y}u_{2}^{*}}} \right)\ {c}}}} - {\frac{1}{G}{\int_{C}{\left( {{u_{1}p_{x}^{*}} + {u_{2}p_{y}^{*}}} \right)\ {c}}}}}} & (1059) \end{matrix}$

The boundary term R of this equation is expressed by eq. (60) as follows:

$\begin{matrix} {R \equiv {\frac{1}{G}{\sum\limits_{i}{\int_{C}^{\;}{\left( {{p_{i}u_{i}^{*}} - {p_{i}^{*}u_{i}}} \right)\ {c}}}}}} & \underset{({Aforementioned})}{(60)} \end{matrix}$

According to the above-described result, partial integration of the left side of eq. (58) gives:

$\begin{matrix} {{\sum\limits_{i}{\sum\limits_{j}{\int_{S}^{\;}{L_{ij}{u_{j} \cdot u_{i}^{*}}{s}}}}} = {R + {\sum\limits_{i}{\sum\limits_{j}{\int_{S}^{\;}{{u_{j} \cdot L_{ij}^{*}}u_{i}^{*}{s}}}}}}} & \underset{({Aforementioned})}{(59)} \end{matrix}$

Incidentally, in order to express the boundary term R of eq. (1059) with polar coordinate components, using eqs. (971), (972), we prepare:

[Formula 464]

u ₁=cos θ·u _(r)−sin θ·u _(θ)  (1060)

u ₂=sin θ·u _(r)+cos θ·u _(θ)  (1061)

and

u ₁*=cos θ·u _(r)*−sin θ·u _(θ)*  (1062)

u ₂*=sin θ·u _(r)*+cos θ·u _(θ)*.  (1063)

Similarly, using eqs. (975), (976), we prepare:

p _(x)=cos θ·p _(r)−sin θ·p _(θ)  (1064)

p _(y)=sin θ·p _(r)+cos θ·p _(θ)  (1065)

and

p _(x)*=cos θ·p _(r)*−sin θ·p _(θ)  (1066)

p _(y)*=sin θ·p _(r)*+cos θ·p _(θ)*  (1067)

Substituting eqs. (1060) to (1067) into eq. (1059) gives:

$\begin{matrix} {R \equiv {{\frac{1}{G}{\int_{C}^{\;}{\left( {{p_{r} \cdot u_{r}^{*}} + {p_{\theta} \cdot u_{\theta}^{*}}} \right)\ {c}}}} - {\frac{1}{G}{\int_{C}^{\;}{\left( {{u_{r} \cdot p_{r}^{*}} + {u_{\theta} \cdot p_{\theta}^{*}}} \right)\ {c}}}}}} & (1068) \end{matrix}$

This equation is a boundary term written with polar coordinate components.

According to a condition such that the boundary term R is zero, the adjoint boundary condition is settled. The following description shows an example of this, in which a radius r of an outer edge of a ring is assumed to satisfy r=R, and a radius r of an inner edge thereof is assumed to satisfy r=γR. This is complicating as the same signs as those of the boundary term R, but the meanings of these are considerably different, and therefore it is possible to distinguish these.

[Exemplary Non-Self-Adjoint Boundary Condition 1]

Regarding a ring, a boundary condition such that the surface force is zero and the displacement is also zero on the inner edge (r=γR) is expressed by eqs. (1018), (1019), and we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 465} \right\rbrack & \; \\ \left. \left. \begin{matrix} {\left. u_{r} \right|_{r = {\gamma \; R}} = 0} \\ {\left. u_{\theta} \right|_{r = {\gamma \; R}} = 0} \\ {\left. p_{r} \right|_{r = {\gamma \; R}} = 0} \\ {\left. p_{\theta} \right|_{r = {\gamma \; R}} = 0} \end{matrix} \right\}\Leftrightarrow\left\{ \begin{matrix} {\left. u_{x} \right|_{r = {\gamma \; R}} = 0} \\ {\left. u_{y} \right|_{r = {\gamma \; R}} = 0} \\ {\left. p_{x} \right|_{r = {\gamma \; R}} = 0} \\ {\left. p_{y} \right|_{r = {\gamma \; R}} = 0} \end{matrix} \right. \right. & (1069) \end{matrix}$

In this case, a condition such that the boundary term R of eq. (60) is zero is given as:

$\begin{matrix} \left. \left. \begin{matrix} {\left. u_{r}^{*} \right|_{r = R} = 0} \\ {\left. u_{\theta}^{*} \right|_{r = R} = 0} \\ {\left. p_{r}^{*} \right|_{r = R} = 0} \\ {\left. p_{\theta}^{*} \right|_{r = R} = 0} \end{matrix} \right\}\Leftrightarrow\left\{ \begin{matrix} {\left. u_{x}^{*} \right|_{r = R} = 0} \\ {\left. u_{y}^{*} \right|_{r = R} = 0} \\ {\left. p_{x}^{*} \right|_{r = R} = 0} \\ {\left. p_{y}^{*} \right|_{r = R} = 0} \end{matrix} \right. \right. & (1070) \end{matrix}$

This is a condition such that on the outer edge (r=R), the dual surface forces p_(x)*, p_(y)* are zero and the dual displacements u_(x)*, u_(y)* are also zero, and this condition is the adjoint boundary condition. As the condition of eq. (1069) and the condition of eq. (1070) are different, this is a non-self-adjoint boundary condition.

It should be noted that the condition equations (1018), (1019) coincide with eq. (1069). In other words, the condition given in Section 11.4.9 is a non-self-adjoint boundary condition.

[Exemplary Self-Adjoint Boundary Condition 1]

Regarding a ring, a boundary condition such that the displacement is zero on the inner edge (r=γR) and the displacement is also zero on the outer edge (r=R) is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 466} \right\rbrack & \; \\ \left. \left. \begin{matrix} {\left. u_{r} \right|_{r = {\gamma \; R}} = 0} \\ {\left. u_{\theta} \right|_{r = {\gamma \; R}} = 0} \\ {\left. u_{r} \right|_{r = R} = 0} \\ {\left. u_{\theta} \right|_{r = R} = 0} \end{matrix} \right\}\Leftrightarrow\left\{ \begin{matrix} {\left. u_{x} \right|_{r = {\gamma \; R}} = 0} \\ {\left. u_{y} \right|_{r = {\gamma \; R}} = 0} \\ {\left. u_{x} \right|_{r = R} = 0} \\ {\left. u_{y} \right|_{r = R} = 0} \end{matrix} \right. \right. & (1071) \end{matrix}$

In this case, a condition such that the boundary term R of eq. (60) is zero is obtained as follows:

$\begin{matrix} \left. \left. \begin{matrix} {\left. u_{r}^{*} \right|_{r = {\gamma \; R}} = 0} \\ {\left. u_{\theta}^{*} \right|_{r = {\gamma \; R}} = 0} \\ {\left. u_{r}^{*} \right|_{r = R} = 0} \\ {\left. u_{\theta}^{*} \right|_{r = R} = 0} \end{matrix} \right\}\Leftrightarrow\left\{ \begin{matrix} {\left. u_{x}^{*} \right|_{r = {\gamma \; R}} = 0} \\ {\left. u_{y}^{*} \right|_{r = {\gamma \; R}} = 0} \\ {\left. u_{x}^{*} \right|_{r = R} = 0} \\ {\left. u_{y}^{*} \right|_{r = R} = 0} \end{matrix} \right. \right. & (1072) \end{matrix}$

As the condition of eq. (1071) and the condition of eq. (1072) coincide, this is a self-adjoint boundary condition.

11.4.13 Homogenization of Boundary Condition and Boundary Term

An index B is added to a term that satisfies an inhomogeneous boundary condition so as to let the term be u_(Bj), and an index H is added to a term that satisfies a homogeneous boundary condition so as to let the term be u_(Hj). A primal displacement u_(j) is expressed by a sum of these, which is given as:

[Formula 467]

u _(j) ≡u _(Bj) +u _(Hj)  (24) (Aforementioned)

Substituting this equation into eq. (23), we obtain the following simultaneous partial differential equation (40) with a homogeneous boundary condition:

$\begin{matrix} {{\sum\limits_{j}{L_{ij}u_{j}}} = f_{i}} & \underset{({Aforementioned})}{(23)} \\ {{\sum\limits_{j}{L_{ij}u_{Hj}}} = f_{Hi}} & \underset{({Aforementioned})}{(40)} \end{matrix}$

An inner product of this with the function u_(Hi)* satisfying the homogeneous adjoint boundary condition is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 468} \right\rbrack & \; \\ {{\sum\limits_{i}{\sum\limits_{j}{\int_{S}^{\;}{L_{ij}{u_{Hj} \cdot u_{Hi}^{*}}{s}}}}} = {\sum\limits_{i}{\int_{S}^{\;}{{f_{Hi} \cdot u_{Hi}^{*}}{s}}}}} & \underset{({Aforementioned})}{(61)} \end{matrix}$

Then, partial integration of the left side of the equation (61) gives the following.

-   -   When |i=1, j=1|, using eqs. (1029), (1030), we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 469} \right\rbrack} & \; \\ \begin{matrix} {{\int_{S}{L_{11}{u_{H\; 1} \cdot u_{H\; 1}^{*}}\ {s}}} \equiv {\int_{S}{\left\lbrack {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\rbrack {u_{H\; 1} \cdot u_{H\; 1}^{*}}{s}}}} \\ {= {{\left( {1 + \mu} \right){\int_{S}{{\frac{\partial^{2}u_{H\; 1}}{\partial x^{2}} \cdot u_{H\; 1}^{*}}\ {s}}}} + {\int_{S}{{\frac{\partial^{2}u_{H\; 1}}{\partial y^{2}} \cdot u_{H\; 1}^{*}}\ {s}}}}} \\ {= {{\int_{C}{\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{H\; 1}}{\partial x}} + {n_{y}\frac{\partial u_{H\; 1}}{\partial y}}} \right\} u_{H\; 1}^{*}\ {c}}} -}} \\ {{{\int_{C}{u_{H\; 1}\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{H\; 1}^{*}}{\partial x}} + {n_{y}\frac{\partial u_{H\; 1}^{*}}{\partial y}}} \right\} \ {c}}} +}} \\ {{\int_{S}{{u_{H\; 1} \cdot \left\lbrack {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\rbrack}u_{H\; 1}^{*}{s}}}} \end{matrix} & (1073) \end{matrix}$

Let the boundary term of the right side of this equation be R_(H11), let the differential operator of the same be L₁₁*, and they are given as:

$\begin{matrix} {\begin{matrix} {R_{H\; 11} \equiv {{\int_{C}{\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{H\; 1}}{\partial x}} + {n_{y}\frac{\partial u_{H\; 1}}{\partial y}}} \right\} u_{H\; 1}^{*}\ {c}}} -}} \\ {{\int_{C}{u_{H\; 1}\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{H\; 1}^{*}}{\partial x}} + {n_{y}\frac{\partial u_{H\; 1}^{*}}{\partial y}}} \right\} \ {c}}}} \end{matrix}{L_{11}^{*} \equiv {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}}}} & (1074) \end{matrix}$

L₁₁* represents an adjoint differential operator shown in eq. (16).

-   -   When |i=1, j=2|, using eq. (1031), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 470} \right\rbrack & \; \\ \begin{matrix} {{\int_{S}{L_{12}{u_{H\; 2} \cdot u_{H\; 1}^{*}}\ {s}}} \equiv {\int_{S}{\left\lbrack {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \right\rbrack {u_{H\; 2} \cdot u_{H\; 1}^{*}}\ {s}}}} \\ {= {{\int_{C}{{\mu\left( {{n_{x}\frac{\partial u_{H\; 2}}{\partial y}u_{H\; 1}^{*}} - {n_{y}u_{H\; 2}\frac{\partial u_{H\; 1}^{*}}{\partial x}}} \right)}\ {c}}} +}} \\ {{\int_{S}{{u_{H\; 2} \cdot \left\lbrack {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \right\rbrack}u_{H\; 1}^{*}{s}}}} \end{matrix} & (1075) \end{matrix}$

Let the boundary term of the right side of this equation be R_(H12), let the differential operator of the same be L₁₂*, and they are given as:

$\begin{matrix} {{R_{H\; 12} \equiv {\int_{C}{{\mu \left( {{n_{x}\frac{\partial u_{H\; 2}}{\partial y}u_{H\; 1}^{*}} - {n_{y}u_{H\; 2}\frac{\partial u_{H\; 1}^{*}}{\partial x}}} \right)}\ {c}}}}{L_{12}^{*} \equiv {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}}}} & (1076) \end{matrix}$

L₁₂* is an adjoint differential operator given in eq. (16).

-   -   When |i=2, j=1|, using eq. (1032), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 471} \right\rbrack & \; \\ \begin{matrix} {{\int_{S}{L_{21}{u_{H\; 1} \cdot u_{H\; 2}^{*}}{s}}} \equiv {\int_{S}{\left\lbrack {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \right\rbrack {u_{H\; 1} \cdot u_{H\; 2}^{*}}{s}}}} \\ {= {{\int_{C}{{\mu\left( {{n_{y}\frac{\partial u_{H\; 1}}{\partial x}u_{H\; 2}^{*}} - {n_{x}u_{H\; 1}\frac{\partial u_{H\; 2}^{*}}{\partial y}}} \right)}\ {c}}} +}} \\ {{\int_{S}{{u_{H\; 1} \cdot \left\lbrack {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}} \right\rbrack}u_{H\; 2}^{*}{s}}}} \end{matrix} & (1077) \end{matrix}$

Let the boundary term of the right side of this equation be R_(H21), let the differential operator of the same be L₂₁*, and they are given as:

$\begin{matrix} {{R_{H\; 21} \equiv {\int_{C}{{\mu\left( {{n_{y}\frac{\partial u_{H\; 1}}{\partial x}u_{H2}^{*}} - {n_{x}u_{H\; 1}\frac{\partial u_{H\; 2}^{*}}{\partial y}}} \right)}\ {c}}}}{L_{12}^{*} \equiv {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}}}} & (1078) \end{matrix}$

L₂₁* is an adjoint differential operator given in eq. (16).

-   -   When |i=2, j=2|, using eqs. (1029), (1030), we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 472} \right\rbrack} & \; \\ \begin{matrix} {{\int_{S}^{\;}{L_{22}{u_{H\; 2} \cdot u_{H\; 2}^{*}}\ {s}}} \equiv {\int_{S}^{\;}{\left\lbrack {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\rbrack {u_{H\; 2} \cdot u_{H\; 2}^{*}}\ {s}}}} \\ {= {{\int_{S}^{\;}{{\frac{\partial^{2}u_{H\; 2}}{\partial x^{2}} \cdot u_{H\; 2}^{*}}\ {s}}} + {\left( {1 + \mu} \right){\int_{S}^{\;}{{\frac{\partial^{2}u_{H\; 2}}{\partial y^{2}} \cdot u_{H\; 2}^{*}}\ {s}}}}}} \\ {= {{\int_{C}^{\;}{\left\{ {{n_{x}\frac{\partial u_{H\; 2}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{H\; 2}}{\partial y}}} \right\} u_{H\; 2}^{*}\ {c}}} -}} \\ {{{\int_{C}^{\;}{u_{H\; 2}\left\{ {{n_{x}\frac{\partial u_{H\; 2}^{*}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{H\; 2}^{*}}{\partial y}}} \right\} \ {c}}} +}} \\ {{\int_{S}^{\;}{{u_{H\; 2} \cdot \left\lbrack {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\rbrack}u_{H\; 2}^{*}\ {s}}}} \end{matrix} & (1079) \end{matrix}$

Let the boundary term of the right side of this equation be R_(H22), let the differential operator of the same be L₂₂*, and they are given as:

$\begin{matrix} {{R_{H\; 22} \equiv {{\int_{C}^{\;}{\left\{ {{n_{x}\frac{\partial u_{H\; 2}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{H\; 2}}{\partial y}}} \right\} u_{H\; 2}^{*}\ {c}}} - {\int_{C}^{\;}{u_{H\; 2}\left\{ {{n_{x}\frac{\partial u_{H\; 2}^{*}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{H\; 2}^{*}}{\partial y}}} \right\} \ {c}}}}}\mspace{20mu} {L_{22}^{*} \equiv {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}}}} & (1080) \end{matrix}$

L₂₂* is an adjoint differential operator given in eq. (16).

In the case where the following is satisfied as in the present case, the operators are referred to as self-adjoint differential operators:

[Formula 473]

L _(ji) *=L _(ij)  (17) (Aforementioned)

Adding the boundary terms and expressing the same as R_(H), we obtain:

$\begin{matrix} {{R_{H} \equiv {R_{H\; 11} + R_{H\; 12} + R_{H\; 21} + R_{H\; 22}}} = {{\int_{C}^{\;}{\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{H\; 1}}{\partial x}} + {n_{y}\frac{\partial u_{H\; 1}}{\partial y}} + {\mu \; n_{x}\frac{\partial u_{H\; 2}}{\partial y}}} \right\} u_{H\; 1}^{*}\ {c}}} + {\int_{C}^{\;}{\left\{ {{n_{x}\frac{\partial u_{H\; 2}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{H\; 2}}{\partial y}} + {\mu \; n_{y}\frac{\partial u_{H\; 1}}{\partial x}}} \right\} u_{H\; 2}^{*}\ {c}}} - {\int_{C}^{\;}{u_{H\; 1}\left\{ {{\left( {1 + \mu} \right)n_{x}\frac{\partial u_{H\; 1}^{*}}{\partial x}} + {n_{y}\frac{\partial u_{H\; 1}^{*}}{\partial y}} + {\mu \; n_{x}\frac{\partial u_{H\; 2}^{*}}{\partial y}}} \right\} \ {c}}} - {\int_{C}^{\;}{u_{H\; 2}\left\{ {{n_{x}\frac{\partial u_{H\; 2}^{*}}{\partial x}} + {\left( {1 + \mu} \right)n_{y}\frac{\partial u_{H\; 2}^{*}}{\partial y}} + {\mu \; n_{y}\frac{\partial u_{H\; 1}^{*}}{\partial x}}} \right\} \ {c}}}}} & (1081) \end{matrix}$

Transforming this equation and utilizing the relationship of boundary integration expressed by eq. (1034), we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 474} \right\rbrack} & \; \\ {R_{H} \equiv {{\int_{C}^{\;}{\left\lbrack {{n_{x}\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{H\; 1}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial u_{H\; 2}}{\partial y}}} \right\}} + {n_{y}\left( {\frac{\partial u_{H\; 2}}{\partial x} + \frac{\partial u_{H\; 1}}{\partial y}} \right)}} \right\rbrack u_{H\; 1}^{*}\ {c}}} + {\int_{C}^{\;}{\left\lbrack {{n_{x}\left( {\frac{\partial u_{H\; 2}}{\partial x} + \frac{\partial u_{H\; 1}}{\partial y}} \right)} + {n_{y}\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{H\; 2}}{\partial y}} + {\left( {\mu - 1} \right)\frac{\partial u_{H\; 1}}{\partial x}}} \right\}}} \right\rbrack u_{H\; 2}^{*}\ {c}}} - {\int_{C}^{\;}{{u_{H\; 1}\left\lbrack {{n_{x}\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{H\; 1}^{*}}{\partial x}} + {\left( {\mu - 1} \right)\frac{\partial u_{H\; 2}^{*}}{\partial y}}} \right)} + {n_{y}\left( {\frac{\partial u_{H\; 2}^{*}}{\partial x} + \frac{\partial u_{H\; 1}^{*}}{\partial y}} \right)}} \right\rbrack}\ {c}}} - {\int_{C}^{\;}{{u_{H\; 2}\left\lbrack {{n_{x}\left( {\frac{\partial u_{H\; 2}^{*}}{\partial x} + \frac{\partial u_{H\; 1}^{*}}{\partial y}} \right)} + {n_{y}\left\{ {{\left( {\mu + 1} \right)\frac{\partial u_{H\; 2}^{*}}{\partial y}} + {\left( {\mu - 1} \right)\frac{\partial u_{H\; 1}^{*}}{\partial x}}} \right\}}} \right\rbrack}\ {c}}}}} & (1082) \end{matrix}$

Using eqs. (27), (30), (33) and eqs. (45), (48), (51), we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 475} \right\rbrack} & \; \\ {R_{H} \equiv {{\frac{1}{G}{\int_{C}^{\;}{\left( {{n_{x}\sigma_{Hx}} + {n_{y}{\tau \ }_{Hyx}}} \right)u_{H\; 1}^{*}{c}}}} + {\frac{1}{G}{\int_{C}^{\;}{\left( {{n_{x}\tau_{Hxy}} + {n_{y}\sigma_{Hy}}} \right)u_{H\; 2}^{*}\ {c}}}} - {\frac{1}{G}{\int_{C}^{\;}{{u_{H\; 1}\left( {{n_{x}\sigma_{Hx}^{*}} + {n_{y}\tau_{Hyx}^{*}}} \right)}\ {c}}}} - {\frac{1}{G}{\int_{C}^{\;}{{u_{H\; 2}\left( {{n_{x}\tau_{Hxy}^{*}} + {n_{y}\sigma_{Hy}^{*}}} \right)}\ {c}}}}}} & (1083) \end{matrix}$

Using eqs. (36), (39) and eqs. (54), (57), that is, Cauchy's formula, we obtain:

$\begin{matrix} {R_{H} \equiv {{\frac{1}{G}{\int_{C}^{\;}{\left( {{p_{Hx}u_{H\; 1}^{*}} + {p_{Hy}u_{H\; 2}^{*}}} \right)\ {c}}}} - {\frac{1}{G}{\int_{C}^{\;}{\left( {{u_{H\; 1}p_{Hx}^{*}} + {u_{H\; 2}p_{Hy}^{*}}} \right)\ {c}}}}}} & (1084) \end{matrix}$

The boundary term R_(H) of this equation is expressed by eq. (63) as follows:

$\begin{matrix} {R_{H} \equiv {\frac{1}{G}{\sum\limits_{i}^{\;}\; {\int_{C}^{\;}{\left( {{p_{Hi}u_{Hi}^{*}} - {p_{Hi}^{*}u_{Hi}}} \right)\ {c}}}}}} & \underset{({Aforementioned})}{(63)} \end{matrix}$

In order that this is expressed with polar coordinate components, this equation is transformed in the same manner as that done when eq. (1068) is obtained, and is given as:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 476} \right\rbrack} & \; \\ {R_{H} \equiv {{\frac{1}{G}{\int_{C}^{\;}{\left( {{p_{Hr} \cdot u_{Hr}^{*}} + {p_{H\; \theta} \cdot u_{H\; \theta}^{*}}} \right)\ {c}}}} - {\frac{1}{G}{\int_{C}^{\;}{\left( {{u_{Hr} \cdot p_{Hr}^{*}} + {u_{H\; \theta} \cdot p_{H\; \theta}^{*}}} \right)\ {c}}}}}} & (1085) \end{matrix}$

According to the results of the above-described operation, partial integration of the left side of the equation (61) gives:

$\begin{matrix} {{\sum\limits_{i}^{\;}{\sum\limits_{j}^{\;}{\int_{S}^{\;}{L_{ij}{u_{Hj} \cdot u_{Hi}^{*}}\ {s}}}}} = {R_{H} + {\sum\limits_{i}^{\;}{\sum\limits_{j}^{\;}{\int_{S}^{\;}{{u_{Hj} \cdot L_{ij}^{*}}u_{Hi}^{*}\ {s}}}}}}} & \underset{({Aforementioned})}{(62)} \end{matrix}$

11.4.14 Getting Equations on Each Function from the Simultaneous Eigenvalue Problem

The outer radius R is assumed to be the characteristic length, and the weight constant is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 477} \right\rbrack & \; \\ {w_{1} = {w_{2} = {w \equiv \frac{1}{R^{2}}}}} & (1086) \end{matrix}$

Then, details of the following primal simultaneous differential equation (105) are given as eqs. (1087) and (1088):

$\begin{matrix} {{\sum\limits_{j}^{\;}\; {L_{ij}\varphi_{j}}} = {\lambda \; w_{i}\varphi_{i}^{*}}} & \underset{({Aforementioned})}{(105)} \\ {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} \varphi_{x}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{y}}} = {\lambda \; w\; \varphi_{x}^{*}}} & (1087) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{x}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} \varphi_{y}}} = {\lambda \; w\; \varphi_{y}^{*}}} & (1088) \end{matrix}$

On the other hand, details of the following dual simultaneous differential equation (106) are given as eqs. (1089) and (1090):

$\begin{matrix} {{\sum\limits_{j}^{\;}\; {L_{ji}^{*}\varphi_{j}^{*}}} = {\lambda \; w_{i}\varphi_{i}}} & \underset{({Aforementioned})}{(106)} \\ {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} \varphi_{x}^{*}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{y}^{*}}} = {\lambda \; w\; \varphi_{x}}} & (1089) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{x}^{*}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} \varphi_{y}^{*}}} = {\lambda \; w\; \varphi_{y}}} & (1090) \end{matrix}$

The primal simultaneous eigenvalue problem is as expressed by eq. (98):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 478} \right\rbrack & \; \\ {{\sum\limits_{j}^{\;}\; {\sum\limits_{k}^{\;}\; {L_{ji}^{*}\frac{1}{w_{j}}L_{jk}\varphi_{k}}}} = {\lambda^{2}w_{i}\varphi_{i}}} & \underset{({Aforementioned})}{(98)} \end{matrix}$

Transposing the weight constant w according to eq. (1086), we obtain:

$\begin{matrix} {{\sum\limits_{j}^{\;}\; {\sum\limits_{k}^{\;}\; {L_{ji}^{*}\frac{1}{w_{j}}L_{jk}L_{jk}\varphi_{k}}}} = {\lambda^{2}w^{2}\varphi_{i}}} & (1091) \end{matrix}$

Using differential operators of eqs. (15), (16), we calculate the left side of this equation, and obtain:

$\begin{matrix} {{{\left\lbrack {\left( {{\nabla^{2}{+ \mu}}\frac{\partial^{2}}{\partial x^{2}}} \right)^{2} + {\mu^{2}\frac{\partial^{4}}{{\partial x^{2}}{\partial y^{2}}}}} \right\rbrack \varphi_{x}} + {\left\lbrack {{\mu \left( {2 + \mu} \right)}\frac{\partial^{2}}{{\partial x}{\partial y}}\nabla^{2}} \right\rbrack \varphi_{y}}} = {\lambda^{2}w^{2}\varphi_{x}}} & (1092) \\ {{{\left\lbrack {{\mu \left( {2 + \mu} \right)}\frac{\partial^{2}}{{\partial x}{\partial y}}\nabla^{2}} \right\rbrack \varphi_{x}} + {\left\lbrack {\left( {{\nabla^{2}{+ \mu}}\frac{\partial^{2}}{\partial y^{2}}} \right)^{2} + {\mu^{2}\frac{\partial^{4}}{{\partial x^{2}}{\partial y^{2}}}}} \right\rbrack \varphi_{y}}} = {\lambda^{2}w^{2}\varphi_{y}}} & (1093) \end{matrix}$

Obtaining equations on each function from these simultaneous equations and rearranging the same, both of φ_(x), and φ_(y) become function φ. that satisfy the following differential equation:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 479} \right\rbrack & \; \\ {{\left( {{\nabla^{4}{- \lambda^{2}}}w^{2}} \right)\left\{ {\nabla^{4}{- \left( \frac{\lambda \; w}{1 + \mu} \right)^{2}}} \right\} \varphi} = 0} & (1094) \end{matrix}$

On the other hand, the dual simultaneous eigenvalue problem is expressed by the following eq. (99):

$\begin{matrix} {{\sum\limits_{j}\; {\sum\limits_{k}\; {L_{ij}\frac{1}{w_{j}}L_{kj}^{*}\varphi_{k}^{*}}}} = {\lambda^{2}w_{i}\varphi_{i}^{*}}} & \underset{({Aforementioned})}{(99)} \end{matrix}$

Similarly this is transformed to:

$\begin{matrix} {{\sum\limits_{j}\; {\sum\limits_{k}\; {L_{ij}L_{kj}^{*}\varphi_{k}^{*}}}} = {\lambda^{2}w^{2}\varphi_{i}^{*}}} & (1095) \end{matrix}$

Calculating the left side of this equation by using the differential operators of eqs. (15), (16), we obtain an equation composed of the same operators as those in eqs. (1092), (1093) as follows:

$\begin{matrix} {{{\left\lbrack {\left( {{\nabla^{2}{+ \mu}}\frac{\partial^{2}}{\partial x^{2}}} \right)^{2} + {\mu^{2}\frac{\partial^{4}}{{\partial x^{2}}{\partial y^{2}}}}} \right\rbrack \varphi_{x}^{*}} + {\left\lbrack {{\mu \left( {2 + \mu} \right)}\frac{\partial^{2}}{{\partial x}{\partial y}}\nabla^{2}} \right\rbrack \varphi_{y}^{*}}} = {\lambda^{2}w^{2}\varphi_{x}^{*}}} & (1096) \\ {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 480} \right\rbrack} & \; \\ {{{\left\lbrack {{\mu \left( {2 + \mu} \right)}\frac{\partial^{2}}{{\partial x}{\partial y}}\nabla^{2}} \right\rbrack \varphi_{x}^{*}} + {\left\lbrack {\left( {{\nabla^{2}{+ \mu}}\frac{\partial^{2}}{\partial y^{2}}} \right)^{2} + {\mu^{2}\frac{\partial^{4}}{{\partial x^{2}}{\partial y^{2}}}}} \right\rbrack \varphi_{y}^{*}}} = {\lambda^{2}w^{2}\varphi_{y}^{*}}} & (1097) \end{matrix}$

Obtaining equations on each function from these simultaneous equations and rearranging the same, both of φ_(x)*,φ_(y)* become function φ* that satisfy the following differential equation:

$\begin{matrix} {{\left( {{\nabla^{4}{- \lambda^{2}}}w^{2}} \right)\left\{ {\nabla^{4}{- \left( \frac{\lambda \; w}{1 + \mu} \right)^{2}}} \right\} \varphi^{*}} = 0} & (1098) \end{matrix}$

11.4.15 Eigenfunction set

Since eqs. (1094) and (1098) are the same differential equation, consequently, we find that φ_(x), φ_(y) and φ_(x)*, φ_(y)* are composed of the same function set respectively. Therefore, focusing on solving eq. (1094), we give:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 481} \right\rbrack & \; \\ {\omega_{1} \equiv \sqrt{\lambda \; w}} & (1099) \\ {\omega_{2} \equiv \sqrt{\frac{\lambda \; w}{1 + \mu}}} & (1100) \end{matrix}$

Then, the differential equation (1094) is transformed to:

(∇⁴−ω₁ ⁴)(∇⁴−ω₂ ⁴)φ=0  (1101)

Factorizing the same, we obtain:

(∇²+ω₁ ²)(∇²−ω₁ ²)(∇²+ω₂ ²)(∇²−ω₂ ²)φ=0  (1102)

Expressing the weight constant iv by eq. (1086), we express ω₁,ω₂ as:

$\begin{matrix} {\omega_{1} \equiv {\frac{1}{R}\sqrt{\lambda}}} & (1103) \\ {\omega_{2} \equiv {\frac{1}{R}\sqrt{\frac{\lambda}{1 + \mu}}} \equiv {\frac{1}{\sqrt{1 + \mu}}\omega_{1}}} & (1104) \end{matrix}$

Incidentally, regarding the function of θ, with periodicity of 2π being expected, let m represent an integer, and a solution thereof is to be determined by separation of variables. Then, a solution of the differential equation (1105) is given by eq. (1106):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 482} \right\rbrack & \; \\ {{\left( {\nabla^{2}{+ \omega_{1}^{2}}} \right)\varphi} = 0} & (1105) \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {J_{m}\left( {\omega_{1}r} \right)} \\ {Y_{m}\left( {\omega_{1}r} \right)} \end{pmatrix}}} & (1106) \end{matrix}$

Here, J_(m) and Y_(m) are Bessel functions of the first kind and the same of the second kind respectively (Bessel function of the first (second) kind). Similarly, a solution of the differential equation (1107) is given by eq. (1108):

$\begin{matrix} {{\left( {\nabla^{2}{+ \omega_{1}^{2}}} \right)\varphi} = 0} & (1107) \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {I_{m}\left( {\omega_{1}r} \right)} \\ {K_{m}\left( {\omega_{1}r} \right)} \end{pmatrix}}} & (1108) \end{matrix}$

Here, I_(m) and K_(m) are modified Bessel functions of the first kind and the same of the second kind respectively (modified Bessel function of the first (second) kind).

Similarly, a solution of the differential equation (1109) is given by eq. (1110):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 483} \right\rbrack & \; \\ {{\left( {\nabla^{2}{+ \omega_{2}^{2}}} \right)\varphi} = 0} & (1109) \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {J_{m}\left( {\omega_{2}r} \right)} \\ {Y_{m}\left( {\omega_{2}r} \right)} \end{pmatrix}}} & (1110) \end{matrix}$

A solution of the differential equation (1111) is given by eq. (1112):

$\begin{matrix} {{\left( {\nabla^{2}{- \omega_{2}^{2}}} \right)\varphi} = 0} & (1111) \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {I_{m}\left( {\omega_{2}r} \right)} \\ {K_{m}\left( {\omega_{2}r} \right)} \end{pmatrix}}} & (1112) \end{matrix}$

11.4.16 Combination of Solutions Satisfying Primal Simultaneous Eigenvalue Problem

Transforming eqs. (1092), (1093) of the primal simultaneous eigenvalue problem, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 484} \right\rbrack & \; \\ {{{\left\lbrack {{\nabla^{4}{- \lambda^{2}}}w^{2}} \right\rbrack \varphi_{x}} + {{\mu \left( {2 + \mu} \right)}{\nabla^{2}\frac{\partial}{\partial x}}\left( {{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} \right)}} = 0} & (1113) \\ {{{\left\lbrack {{\nabla^{4}{- \lambda^{2}}}w^{2}} \right\rbrack \varphi_{y}} + {{\mu \left( {2 + \mu} \right)}{\nabla^{2}\frac{\partial}{\partial y}}\left( {{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} \right)}} = 0} & (1114) \end{matrix}$

Using ω₁ of eq. (1099), we obtain:

$\begin{matrix} {{{\left\lbrack {\nabla^{4}{- w_{1}^{4}}} \right\rbrack \varphi_{x}} + {{\mu \left( {2 + \mu} \right)}{\nabla^{2}\frac{\partial}{\partial x}}\left( {{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} \right)}} = 0} & (1115) \\ {{{\left\lbrack {\nabla^{4}{- w_{1}^{4}}} \right\rbrack \varphi_{y}} + {{\mu \left( {2 + \mu} \right)}{\nabla^{2}\frac{\partial}{\partial y}}\left( {{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} \right)}} = 0} & (1116) \end{matrix}$

In the case where the solutions φ_(x), φ_(y) are selected from a function system that satisfies eq. (1105) or eq. (1107), both of φ_(x), φ_(y) satisfy:

[Formula 485]

(∇⁴−ω₁ ⁴)φ=0  (1117)

Therefore, according to eqs. (1115), (1116), the combination of φ_(x),φ_(y) has to satisfy:

$\begin{matrix} {{\frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} \right)} = 0} & (1118) \\ {{\frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} \right)} = 0} & (1119) \end{matrix}$

This requires that the terms in the parenthesis are constants. However, since the solution formats of φ_(x), φ_(y) are eqs. (1106), (1108), they do not take a constant other than zero. Therefore, as an equation that the combination of φ_(x), φ_(y) should satisfy, we obtain the following equation:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 486} \right\rbrack & \; \\ {{{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} = 0} & (1120) \end{matrix}$

On the other hand, eqs. (1092), (1093) of the primal simultaneous eigenvalue problem can be transformed to:

$\begin{matrix} {{{\left\lbrack {\nabla^{4}{- \left( \frac{\lambda \; w}{1 + \mu} \right)^{2}}} \right\rbrack \varphi_{x}} - {\frac{\mu \left( {2 + \mu} \right)}{1 + \mu}{\nabla^{2}\frac{\partial}{\partial y}}\left( {{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} \right)}} = 0} & (1121) \\ {{{\left\lbrack {\nabla^{4}{- \left( \frac{\lambda \; w}{1 + \mu} \right)^{2}}} \right\rbrack \varphi_{y}} + {\frac{\mu \left( {2 + \mu} \right)}{1 + \mu}{\nabla^{2}\frac{\partial}{\partial x}}\left( {{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} \right)}} = 0} & (1122) \end{matrix}$

Using ω₂ of eq. (1100), we obtain:

$\begin{matrix} {{{\left\lbrack {\nabla^{4}{- \omega_{2}^{4}}} \right\rbrack \varphi_{x}} - {\frac{\mu \left( {2 + \mu} \right)}{1 + \mu}{\nabla^{2}\frac{\partial}{\partial y}}\left( {{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} \right)}} = 0} & (1123) \\ {{{\left\lbrack {\nabla^{4}{- \omega_{2}^{4}}} \right\rbrack \varphi_{y}} + {\frac{\mu \left( {2 + \mu} \right)}{1 + \mu}{\nabla^{2}\frac{\partial}{\partial x}}\left( {{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} \right)}} = 0} & (1124) \end{matrix}$

In the case where the solutions φ_(x), φ_(y) are selected from a function system that satisfies eq. (1109) or (1111), both of φ_(x), φ_(y) satisfy:

[Formula 487]

(∇⁴−ω₂ ⁴)φ=0  (1125)

Therefore, according to eqs. (1123), (1124), the combination of φ_(x),φ_(y) has to satisfy:

$\begin{matrix} {{\frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} \right)} = 0} & (1126) \\ {{\frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} \right)} = 0} & (1127) \end{matrix}$

This requires that the terms in the parenthesis are constants. However, since the solution formats of φ_(x), φ_(y), are eqs. (1110), (1112), they do not take a constant other than zero. Therefore, as an equation that the combination of φ_(x), φ_(y) should satisfy, we obtain the following equation:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 488} \right\rbrack & \; \\ {{{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} = 0} & (1128) \end{matrix}$

11.4.17 Combination of Solutions Satisfying Dual Simultaneous Eigenvalue Problem

Transforming eqs. (1096), (1097) of the dual simultaneous eigenvalue problem, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 489} \right\rbrack & \; \\ {{{\left\lbrack {{\nabla^{4}{- \lambda^{2}}}w^{2}} \right\rbrack \varphi_{x}^{*}} + {{\mu \left( {2 + \mu} \right)}{\nabla^{2}\frac{\partial}{\partial x}}\left( {{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} \right)}} = 0} & (1129) \\ {{{\left\lbrack {{\nabla^{4}{- \lambda^{2}}}w^{2}} \right\rbrack \varphi_{y}^{*}} + {{\mu \left( {2 + \mu} \right)}{\nabla^{2}\frac{\partial}{\partial y}}\left( {{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} \right)}} = 0} & (1130) \end{matrix}$

Using ω₁ of eq. (1099), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 490} \right\rbrack & \; \\ {{{\left\lbrack {\nabla^{4}{- w_{1}^{4}}} \right\rbrack \varphi_{x}^{*}} + {{\mu \left( {2 + \mu} \right)}{\nabla^{2}\frac{\partial}{\partial x}}\left( {{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} \right)}} = 0} & (1131) \\ {{{\left\lbrack {\nabla^{4}{- w_{1}^{4}}} \right\rbrack \varphi_{y}^{*}} + {{\mu \left( {2 + \mu} \right)}{\nabla^{2}\frac{\partial}{\partial y}}\left( {{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} \right)}} = 0} & (1132) \end{matrix}$

In the case where the solutions φ_(x)*,φ_(y)* are selected from a function system that satisfies eq. (1105) or eq. (1107), both of φ_(x)*,φ_(y)* satisfy:

(∇⁴−ω₁ ⁴)φ*=0  (1133)

Therefore, according to eqs. (1131), (1132), the combination of φ_(x)*,φ_(y)* has to satisfy:

$\begin{matrix} {{\frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} \right)} = 0} & (1134) \\ {{\frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} \right)} = 0} & (1135) \end{matrix}$

This requires that the terms in the parenthesis are constants. However, since the solution formats of φ_(x)*, (Ware eqs. (1106), (1108), they do not take a constant other than zero. Therefore, as an equation that the combination of φ_(x)*, φ_(y) ^(*) should satisfy, we obtain the following equation:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 491} \right\rbrack & \; \\ {{{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} = 0} & (1136) \end{matrix}$

On the other hand, eqs. (1096), (1097) of the dual simultaneous eigenvalue problem can be transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 492} \right\rbrack & \; \\ {{{\left\lbrack {\nabla^{4}{- \left( \frac{\lambda \; w}{1 + \mu} \right)^{2}}} \right\rbrack \varphi_{x}^{*}} - {\frac{\mu \left( {2 + \mu} \right)}{1 + \mu}{\nabla^{2}\frac{\partial}{\partial y}}\left( {{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} \right)}} = 0} & (1137) \\ {{{\left\lbrack {\nabla^{4}{- \left( \frac{\lambda \; w}{1 + \mu} \right)^{2}}} \right\rbrack \varphi_{y}^{*}} + {\frac{\mu \left( {2 + \mu} \right)}{1 + \mu}{\nabla^{2}\frac{\partial}{\partial x}}\left( {{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} \right)}} = 0} & (1138) \end{matrix}$

Using ω₂ of eq. (1100), we obtain:

$\begin{matrix} {{{\left\lbrack {\nabla^{4}{- \omega_{2}^{4}}} \right\rbrack \varphi_{x}^{*}} - {\frac{\mu \left( {2 + \mu} \right)}{1 + \mu}{\nabla^{2}\frac{\partial}{\partial y}}\left( {{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} \right)}} = 0} & (1139) \\ {{{\left\lbrack {\nabla^{4}{- \omega_{2}^{4}}} \right\rbrack \varphi_{y}^{*}} + {\frac{\mu \left( {2 + \mu} \right)}{1 + \mu}{\nabla^{2}\frac{\partial}{\partial x}}\left( {{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} \right)}} = 0} & (1140) \end{matrix}$

In the case where the solutions φ_(x)*, φ_(y)* are selected from a function system that satisfies eq. (1109) or (1111), both of φ_(x)*, φ_(y)* satisfy:

[Formula 493]

(∇⁴−ω₂ ⁴)φ*=0  (1141)

Therefore, according to eqs. (1139), (1140), the combination of φ_(x)*,φ_(y)*; has to satisfy:

$\begin{matrix} {{\frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} \right)} = 0} & (1142) \\ {{\frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} \right)} = 0} & (1143) \end{matrix}$

This requires that the terms in the parenthesis are constants. However, since the solution formats of φ_(x)*, φ_(y)* are eqs. (1110), (1112), they do not take a constant other than zero. Therefore, as an equation that the combination of φ_(x)*, φ_(y)*should satisfy, we obtain the following equation:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 494} \right\rbrack & \; \\ {{{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} = 0} & (1144) \end{matrix}$

11.4.18 Combination of Solutions Satisfying Primal Simultaneous Differential Equations

Transforming eqs. (1087), (1088) of the primal simultaneous differential equations, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 495} \right\rbrack & \; \\ {{{\nabla^{2}\varphi_{x}} + {\mu \frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} \right)}} = {\lambda \; w\; \varphi_{x}^{*}}} & (1145) \\ {{{\nabla^{2}\varphi_{y}} + {\mu \frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} \right)}} = {\lambda \; w\; \varphi_{y}^{*}}} & (1146) \end{matrix}$

Using ω₁ of eq. (1099), we obtain:

$\begin{matrix} {{{\nabla^{2}\varphi_{x}} + {\mu \frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} \right)}} = {w_{1}^{2}\varphi_{x}^{*}}} & (1147) \\ {{{\nabla^{2}\varphi_{y}} + {\mu \frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} \right)}} = {w_{1}^{2}\varphi_{y}^{*}}} & (1148) \end{matrix}$

In the case where the solutions φ_(x), φ_(y) are selected from a function system of the following eq. (1106) satisfying the following eq. (1105), the solutions φ_(x), φ_(y) have to satisfy the following eq. (1120):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 496} \right\rbrack & \; \\ {{\left( {\nabla^{2}{+ \omega_{1}^{2}}} \right)\varphi} = 0} & \underset{({Aforementioned})}{(1105)} \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {J_{m}\left( {\omega_{1}r} \right)} \\ {Y_{m}\left( {\omega_{1}r} \right)} \end{pmatrix}}} & \underset{({Aforementioned})}{(1106)} \\ {{{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} = 0} & \underset{({Aforementioned})}{(1120)} \end{matrix}$

Therefore, eqs. (1147), (1148) are transformed to:

∇²φ_(x) =w ₁ ²φ_(x) *=−w ₁ ²φ_(x)  (1149)

∇²φ_(y) =w ₁ ²φ_(y) *=−w ₁ ²φ_(y)  (1150)

From eqs. (1149) and (1150), we obtain:

φ_(x)*=−φ_(x)  (1151)

φ_(y)*=−φ_(y)  (1152)

Substituting these into the primal simultaneous differential equations (1087), (1088), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 497} \right\rbrack & \; \\ {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} \varphi_{x}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{y}}} = {{- \lambda}\; w\; \varphi_{x}}} & (1153) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{x}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} \varphi_{y}}} = {{- \lambda}\; w\; \varphi_{y}}} & (1154) \end{matrix}$

This indicates that a combination of φ_(x), φ_(y) that is formed with a function system of eq. (1106) and satisfies eq. (1120) satisfies these simultaneous differential equations.

In the case where the solutions φ_(x), φ_(y) are selected from a function system of the following eq. (1108) satisfying the following eq. (1107), the solutions φ_(x), φ_(y) have to satisfy the following eq. (1120):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 498} \right\rbrack & \; \\ {{\left( {\nabla^{2}{- \omega_{1}^{2}}} \right)\varphi} = 0} & \begin{matrix} {\mspace{115mu} (1107)} \\ ({Aforementioned}) \end{matrix} \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {I_{m}\left( {\omega_{1}r} \right)} \\ {K_{m}\left( {\omega_{1}r} \right)} \end{pmatrix}}} & \begin{matrix} {\mspace{115mu} (1108)} \\ ({Aforementioned}) \end{matrix} \\ {{{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} = 0} & \begin{matrix} {\mspace{115mu} (1120)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Therefore, eqs. (1147), (1148) are transformed to:

∇²φ_(x) =w ₁ ²φ_(x) *=w ₁ ²φ_(x)  (1155)

∇²φ_(y) =w ₁ ²φ_(y) *=w ₁ ²φ_(y)  (1156)

From eqs. (1155) and (1156), we obtain:

[Formula 499]

φ_(x)*=φ_(x)  (1157)

φ_(y)*=φ_(y)  (1158)

Substituting these into the primal simultaneous differential equations (1087), (1088), we obtain:

$\begin{matrix} {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} \varphi_{x}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{y}}} = {{+ \lambda}\; w\; \varphi_{x}}} & (1159) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{x}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} \varphi_{y}}} = {{+ \lambda}\; w\; \varphi_{y}}} & (1160) \end{matrix}$

This indicates that a combination of φ_(x), φ_(y) that is formed with a function system of eq. (1108) and satisfies eq. (1120) satisfies these simultaneous differential equations.

On the other hand, eqs. (1087), (1088) of the primal simultaneous differential equations can be transformed also to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 500} \right\rbrack & \; \\ {{{\nabla^{2}\varphi_{x}} - {\frac{\mu}{1 + \mu}\frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} \right)}} = {\frac{\lambda \; w}{1 + \mu}\varphi_{x}^{*}}} & (1161) \\ {{{\nabla^{2}\varphi_{y}} - {\frac{\mu}{1 + \mu}\frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} \right)}} = {\frac{\lambda \; w}{1 + \mu}\varphi_{y}^{*}}} & (1162) \end{matrix}$

Using ω₂ of eq. (1100), we obtain:

$\begin{matrix} {{{\nabla^{2}\varphi_{x}} - {\frac{\mu}{1 + \mu}\frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} \right)}} = {w_{2}^{2}\varphi_{x}^{*}}} & (1163) \\ {{{\nabla^{2}\varphi_{y}} - {\frac{\mu}{1 + \mu}\frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} \right)}} = {w_{2}^{2}\varphi_{y}^{*}}} & (1164) \end{matrix}$

[Formula 501]

In the case where the solutions φ_(x),φ_(y) are selected from a function system of the following eq. (1110) satisfying the following eq. (1109), the solutions φ_(x),φ_(y) have to satisfy the following eq. (1128):

$\begin{matrix} {{\left( {\nabla^{2}{+ \omega_{2}^{2}}} \right)\varphi} = 0} & \begin{matrix} {\mspace{115mu} (1109)} \\ ({Aforementioned}) \end{matrix} \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {J_{m}\left( {\omega_{2}r} \right)} \\ {Y_{m}\left( {\omega_{2}r} \right)} \end{pmatrix}}} & \begin{matrix} {\mspace{115mu} (1110)} \\ ({Aforementioned}) \end{matrix} \\ {{{\frac{\partial}{\partial y}\varphi_{x}} - {\frac{\partial}{\partial x}\varphi_{y}}} = 0} & \begin{matrix} {\mspace{115mu} (1128)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Therefore, eqs. (1163), (1164) are transformed to:

∇²φ_(x) =w ₂ ²φ_(x) *=−w ₂ ²φ_(x)  (1165)

∇²φ_(y) =w ₂ ²φ_(y) *=−w ₂ ²φ_(y)  (1166)

[Formula 502]

From eqs. (1165) and (1166), we obtain:

φ_(x)*=−φ_(x)  (1167)

φ_(y)*=−φ_(y)  (1168)

Substituting these into the primal simultaneous differential equations (1087), (1088), we obtain:

$\begin{matrix} {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} \varphi_{x}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{y}}} = {{- \lambda}\; w\; \varphi_{x}}} & (1169) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{x}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} \varphi_{y}}} = {{- \lambda}\; w\; \varphi_{y}}} & (1170) \end{matrix}$

This indicates that a combination of φ_(x), φ_(y) that is formed with a function system of eq. (1110) and satisfies eq. (1128) satisfies these simultaneous differential equations.

[Formula 503]

In the case where the solutions φ_(x),φ_(y) are selected from a function system of the following eq. (1112) satisfying the following eq. (1111), the solutions φ_(x),φ_(y) have to satisfy the following eq. (1128):

$\begin{matrix} {{\left( {\nabla^{2}{- \omega_{2}^{2}}} \right)\varphi} = 0} & \begin{matrix} {\mspace{115mu} (1111)} \\ ({Aforementioned}) \end{matrix} \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {I_{m}\left( {\omega_{2}r} \right)} \\ {K_{m}\left( {\omega_{2}r} \right)} \end{pmatrix}}} & \begin{matrix} {\mspace{115mu} (1112)} \\ ({Aforementioned}) \end{matrix} \\ {{{\frac{\partial}{\partial y}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} = 0} & \begin{matrix} {\mspace{115mu} (1128)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Therefore, eqs. (1163), (1164) are transformed to:

[Formula 504]

∇²φ_(x) =w ₂ ²φ_(x) *=w ₂ ²φ_(x)  (1171)

∇²φ_(y) =w ₂ ²φ_(y) *=w ₂ ²φ_(y)  (1172)

From eqs. (1171) and (1172), we obtain:

φ_(x)*=φ_(x)  (1173)

φ_(y)*=φ_(y)  (1174)

Substituting these into the primal simultaneous differential equations (1087), (1088), we obtain:

$\begin{matrix} {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} \varphi_{x}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{y}}} = {{+ \lambda}\; w\; \varphi_{x}}} & (1175) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{x}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} \varphi_{y}}} = {{+ \lambda}\; w\; \varphi_{y}}} & (1176) \end{matrix}$

This indicates that a combination of φ_(x), φ_(y) that is formed with a function system of eq. (1112) and satisfies eq. (1128) satisfies these simultaneous differential equations.

11.4.19 Combination of Solutions Satisfying Dual Simultaneous Differential Equations

Transforming eqs. (1089), (1090) of the dual simultaneous differential equations, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 505} \right\rbrack & \; \\ {{{{\nabla^{2}\varphi_{x}^{*}} + {\mu \frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} \right)}} = {\lambda \; w\; \varphi_{x}}}\;} & (1177) \\ {{{\nabla^{2}\varphi_{y}^{*}} + {\mu \frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} \right)}} = {\lambda \; w\; \varphi_{y}}} & (1178) \end{matrix}$

Using ω₁ of eq. (1099), we obtain:

$\begin{matrix} {{{{\nabla^{2}\varphi_{x}^{*}} + {\mu \frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} \right)}} = {w_{1}^{2}\varphi_{x}}}\;} & (1179) \\ \left\lbrack {{Formula}\mspace{14mu} 506} \right\rbrack & \; \\ {{{\nabla^{2}\varphi_{y}^{*}} + {\mu \frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} \right)}} = {w_{1}^{2}\varphi_{y}}} & (1180) \end{matrix}$

In the case where the solutions φ_(x)*, φ_(y)* are selected from a function system of the following eq. (1106) satisfying the following eq. (1105), the solutions φ_(x)*, φ_(y)* have to satisfy the following eq. (1136):

$\begin{matrix} {{\left( {\nabla^{2}{- \omega_{1}^{2}}} \right)\varphi} = 0} & \begin{matrix} {\mspace{115mu} (1105)} \\ ({Aforementioned}) \end{matrix} \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {J_{m}\left( {\omega_{1}r} \right)} \\ {Y_{m}\left( {\omega_{1}r} \right)} \end{pmatrix}}} & \begin{matrix} {\mspace{115mu} (1106)} \\ ({Aforementioned}) \end{matrix} \\ {{{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} = 0} & \begin{matrix} {\mspace{115mu} (1136)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Therefore, eqs. (1179), (1180) are transformed to:

[Formula 507]

∇²φ_(x) *=w ₁ ²φ_(x) =−w ₁ ²φ_(x)*  (1181)

∇²φ_(y) *=w ₁ ²φ_(y) =−w ₁ ²φ_(y)*  (1182)

From eqs. (1181) and (1182), we obtain:

φ_(x)=−φ_(x)*  (1183)

φ_(y)=−φ_(y)*  (1184)

Substituting these into the dual simultaneous differential equations (1089), (1090), we obtain:

$\begin{matrix} {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} \varphi_{x}^{*}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{y}^{*}}} = {{+ \lambda}\; w\; \varphi_{x}^{*}}} & (1185) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{x}^{*}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} \varphi_{y}^{*}}} = {{+ \lambda}\; w\; \varphi_{y}^{*}}} & (1186) \end{matrix}$

This indicates that a combination of φ_(x)*, φ_(y)* that is formed with a function system of eq. (1106) and satisfies eq. (1136) satisfies these simultaneous differential equations.

In the case where the solutions φ_(x)*, φ_(y)* are selected from a function system of the following eq. (1108) satisfying the following eq. (1107), the solutions φ_(x)*, φ_(y)* have to satisfy the following eq. (1136):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 508} \right\rbrack & \; \\ {{\left( {\nabla^{2}{- \omega_{1}^{2}}} \right)\varphi} = 0} & \begin{matrix} {\mspace{115mu} (1107)} \\ ({Aforementioned}) \end{matrix} \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {I_{m}\left( {\omega_{1}r} \right)} \\ {K_{m}\left( {\omega_{1}r} \right)} \end{pmatrix}}} & \begin{matrix} {\mspace{115mu} (1108)} \\ ({Aforementioned}) \end{matrix} \\ {{{\frac{\partial}{\partial x}\varphi_{x}^{*}} + {\frac{\partial}{\partial y}\varphi_{y}^{*}}} = 0} & \begin{matrix} {\mspace{115mu} (1136)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Therefore, eqs. (1179), (1180) are transformed to:

∇²φ_(x) *=w ₁ ²φ_(x) =w ₁ ²φ_(x)*  (1187)

∇²φ_(y) *=w ₁ ²φ_(y) =w ₁ ²φ_(y)*  (1188)

From eqs. (1187) and (1188), we obtain:

φ_(x)=φ_(x)*  (1189)

φ_(y)=φ_(y)*  (1190)

Substituting these into the dual simultaneous differential equations (1089), (1090), we obtain:

$\begin{matrix} {\left\lbrack {{Formula}\mspace{14mu} 509} \right\rbrack \begin{matrix} {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} \varphi_{x}^{*}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{y}^{*}}} = {{+ \lambda}\; w\; \varphi_{x}^{*}}} & (1191) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{x}^{*}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} \varphi_{y}^{*}}} = {{+ \lambda}\; w\; \varphi_{y}^{*}}} & (1186) \end{matrix}} & \; \end{matrix}$

This indicates that a combination of φ_(x)*, φ_(y)* that is formed with a function system of eq. (1108) and satisfies eq. (1136) satisfies these simultaneous differential equations.

On the other hand, eqs. (1089), (1090) of the dual simultaneous differential equations can be transformed to:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 510} \right\rbrack & \; \\ {{{\nabla^{2}\varphi_{x}^{*}} - {\frac{\mu}{1 + \mu}\frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} \right)}} = {\frac{\lambda \; w}{1 + \mu}\varphi_{x}}} & (1193) \\ {{{\nabla^{2}\varphi_{y}^{*}} + {\frac{\mu}{1 + \mu}\frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} \right)}} = {\frac{\lambda \; w}{1 + \mu}\varphi_{y}}} & (1194) \end{matrix}$

Using ω₂ of eq. (1100), we obtain:

$\begin{matrix} {\left\lbrack {{Formula}\mspace{14mu} 511} \right\rbrack \begin{matrix} {{{\nabla^{2}\varphi_{x}^{*}} - {\frac{\mu}{1 + \mu}\frac{\partial}{\partial y}\left( {{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} \right)}} = {w_{2}^{2}\varphi_{x}}} & (1195) \\ {{{\nabla^{2}\varphi_{y}^{*}} + {\frac{\mu}{1 + \mu}\frac{\partial}{\partial x}\left( {{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} \right)}} = {w_{2}^{2}\varphi_{y}}} & (1196) \end{matrix}} & \; \end{matrix}$

In the case where the solutions φ_(x)*,φ_(y)* are selected from a function system of the following eq. (1110) satisfying the following eq. (1109), the solutions φ_(x)*,φ_(y)* have to satisfy the following eq. (1144):

$\begin{matrix} {{\left( {\nabla^{2}{+ \omega_{2}^{2}}} \right)\varphi} = 0} & \begin{matrix} {\mspace{115mu} (1109)} \\ ({Aforementioned}) \end{matrix} \\ \left\lbrack {{Formula}\mspace{14mu} 512} \right\rbrack & \; \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {J_{m}\left( {\omega_{2}r} \right)} \\ {Y_{m}\left( {\omega_{2}r} \right)} \end{pmatrix}}} & \begin{matrix} {\mspace{115mu} (1110)} \\ ({Aforementioned}) \end{matrix} \\ {{{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} = 0} & \begin{matrix} {\mspace{115mu} (1144)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Therefore, eqs. (1195), (1196) are transformed to:

∇²φ_(x) *=w ₂ ²φ_(x) =−w ₂ ²φ_(x)*  (1197)

∇²φ_(y) *=w ₂ ²φ_(y) =−w ₂ ²φ_(y)*  (1198)

From eqs. (1197) and (1198), we obtain:

φ_(x)=−φ_(x)*  (1199)

φ_(y)=φ_(y)*  (1200)

Substituting these into the dual simultaneous differential equations (1089), (1090), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 513} \right\rbrack & \; \\ {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} \varphi_{x}^{*}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{y}^{*}}} = {{- \lambda}\; w\; \varphi_{x}^{*}}} & (1201) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{x}^{*}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} \varphi_{y}^{*}}} = {{- \lambda}\; w\; \varphi_{y}^{*}}} & (1202) \end{matrix}$

This indicates that a combination of φ_(x)*, φ_(y)* that is formed with a function system of eq. (1110) and satisfies eq. (1144) satisfies these simultaneous differential equations.

In the case where the solutions φ_(x)*, φ_(y)* are selected from a function system of the following eq. (1112) satisfying the following eq. (1111), the solutions φ_(x)*, φ_(y)* have to satisfy the following eq. (1144):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 514} \right\rbrack & \; \\ {{\left( {\nabla^{2}{- \omega_{2}^{2}}} \right)\varphi} = 0} & \begin{matrix} {\mspace{115mu} (1111)} \\ ({Aforementioned}) \end{matrix} \\ {{\varphi \left( {r,\theta} \right)} \equiv {\begin{pmatrix} {\sin \left( {m\; \theta} \right)} \\ {\cos \left( {m\; \theta} \right)} \end{pmatrix} \times \begin{pmatrix} {I_{m}\left( {\omega_{2}r} \right)} \\ {K_{m}\left( {\omega_{2}r} \right)} \end{pmatrix}}} & \begin{matrix} {\mspace{115mu} (1112)} \\ ({Aforementioned}) \end{matrix} \\ {{{\frac{\partial}{\partial y}\varphi_{x}^{*}} - {\frac{\partial}{\partial x}\varphi_{y}^{*}}} = 0} & \begin{matrix} {\mspace{115mu} (1144)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Therefore, eqs. (1195), (1196) are transformed to:

[Formula 515]

∇²φ_(x) *=w ₂ ²φ_(x) =w ₂ ²φ_(x)*  (1203)

∇²φ_(y) *=w ₂ ²φ_(y) =w ₂ ²φ_(y)*  (1204)

From eqs. (1203) and (1204), we obtain:

φ_(x)=φ_(x)*  (1205)

φ_(y)=φ_(y)*  (1206)

Substituting these into the dual simultaneous differential equations (1089), (1090), we obtain:

$\begin{matrix} {{{\left\{ {{\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial x^{2}}} + \frac{\partial^{2}}{\partial y^{2}}} \right\} \varphi_{x}^{*}} + {\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{y}^{*}}} = {{+ \lambda}\; w\; \varphi_{x}^{*}}} & (1207) \\ {{{\mu \frac{\partial^{2}}{{\partial x}{\partial y}}\varphi_{x}^{*}} + {\left\{ {\frac{\partial^{2}}{\partial x^{2}} + {\left( {1 + \mu} \right)\frac{\partial^{2}}{\partial y^{2}}}} \right\} \varphi_{y}^{*}}} = {{+ \lambda}\; w\; \varphi_{y}^{*}}} & (1208) \end{matrix}$

This indicates that a combination of φ_(x)*, φ_(y)* that is formed with a function system of eq. (1112) and satisfies eq. (1144) satisfies these simultaneous differential equations.

11.4.20 Appearance of Solution Function

In using the solutions of eqs. (1106), (1108), (1110), and (1112), the functions are classified depending on the integer m, into a category with the integer m of an odd number and a category with the integer m of an even number. Therefore let n be an integer, and the following is given:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 516} \right\rbrack & \; \\ {m \equiv \left\{ \begin{matrix} {2n} & {{even}\mspace{14mu} {number}} \\ {{2n} + 1} & {{odd}\mspace{14mu} {number}} \end{matrix} \right.} & (1209) \end{matrix}$

Then, the formats of the functions are classified into the following four categories:

(1) If a function φ_(ee) is defined as

φ_(ee)(θ)≡cos 2n θ  (1210)

the following equations are established:

φ_(ee)(θ)=φ_(ee)(−θ), φ_(ee)(θ)=φ_(ee)(π−θ)  (1211)

Therefore, the function becomes a function that is vertically symmetric (symmetric with respect to the x axis), and horizontally symmetric (symmetric with respect to the y axis). In other words, the function is even in the x direction, and is even in the y direction. Thus, the index of “ee”, which is an acronym of “even” and “even”, is added.

(2) If a function φ_(oe) is defined as

[Formula 517]

φ_(oe)(θ)≡cos(2n+1)θ,  (1212)

the following equations are established:

φ_(oe)(θ)=φ_(oe)(−θ), φ_(oe)(θ)=−φ_(oe)(π−θ)  (1213)

Therefore, the function becomes a function that is vertically symmetric (symmetric with respect to the x axis), and horizontally anti-symmetric (anti-symmetric with respect to the y axis). In other words, the function is odd in the x direction, and is even in the y direction. Thus, the index of “oe”, which is an acronym of “odd” and “even”, is added.

(3) If a function φ_(oo) is defined as

[Formula 518]

φ_(oo)(θ)≡sin 2nθ,  (1214)

the following equations are established:

φ_(oo)(θ)=φ_(oo)(−θ), φ_(oo)(θ)=−φ_(oo)(π−θ)  (1215)

Therefore, the function becomes a function that is vertically anti-symmetric (anti-symmetric with respect to the x axis), and horizontally anti-symmetric (anti-symmetric with respect to the y axis). In other words, the function is odd in the x direction, and is odd in the y direction. Thus, the index of “oo”, which is an acronym of “odd” and “odd”, is added.

(4) If a function φ_(eo) is defined as,

[Formula 519]

φ_(eo)(θ)≡sin(2n+1)θ,  (1216)

the following equations are established:

φ_(eo)(θ)=−φ_(eo)(−θ), φ_(eo)(θ)=φ_(eo)(π−θ)  (1217)

Therefore, the function becomes a function that is vertical anti-symmetric (anti-symmetric with respect to the x axis) and horizontally symmetric (symmetric with respect to the y axis). In other words, the function is even in the x direction, and is odd in the y direction. Thus, the index of “eo”, which is an acronym of “even” and “odd”, is added.

Next, the following typical deformation modes occur, depending on the above-described combination of the functions having a displacement in the x direction as φx, and a displacement in the y direction as φ_(y):

[SA]

When the displacement in the x direction is expressed by the function φ_(xee) in the form of eq. (1210) and the displacement in the y direction is expressed by the function φ_(yoo) in the form of eq. (1214), the deformation is symmetric with respect to the x axis and asymmetric with respect to the y axis, as shown in FIG. 43. Such a deformation is called a “mode SA”, “SA” being an acronym of “symmetric” and “asymmetric”. The combination φ_(SA) of functions that expresses this deformation is defined as follows, with an index of “SA”:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 520} \right\rbrack & \; \\ {\varphi_{SA} \equiv \left\{ \begin{matrix} \varphi_{xee} \\ \varphi_{yoo} \end{matrix} \right.} & (1218) \end{matrix}$

[AS]

When the displacement in the x direction is expressed by function φ_(xoo) in the form of eq. (1214) and the displacement in the y direction is expressed by the function φ_(yee) in the form of eq. (1210), the deformation is asymmetric with respect to the x axis and symmetric with respect to the y axis, as shown in FIG. 44. Such a deformation is called a “mode AS”, AS being an acronym of “asymmetric” and “symmetric”. The combination φ_(AS) of functions that expresses this deformation is defined as follows, with an index of “AS”:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 521} \right\rbrack & \; \\ {\varphi_{AS} \equiv \left\{ \begin{matrix} \varphi_{xoo} \\ \varphi_{yee} \end{matrix} \right.} & (1219) \end{matrix}$

[SS]

When the displacement in the x direction is expressed by the function φ_(xoe) in the form of eq. (1212), and the displacement in the y direction is expressed by the function φ_(yeo) in the form of eq. (1216), the deformation is symmetric with respect to the x axis and symmetric with respect to the y axis, as shown in FIG. 45. Such a deformation is called a “mode SS”, “SS” being an acronym of “symmetric” and “symmetric”. The combination φ_(ss) of functions that expresses this deformation is defined as follows, with an index of “SS”:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 522} \right\rbrack & \; \\ {\varphi_{SS} \equiv \left\{ \begin{matrix} \varphi_{xoe} \\ \varphi_{yeo} \end{matrix} \right.} & (1220) \end{matrix}$

[AA]

When the displacement in the x direction is expressed by the function φ_(xeo) in the form of eq. (1216) and the displacement in the y direction is expressed by the function φ_(yoe) in the form of eq. (1212), the deformation is asymmetric with respect to the x axis and asymmetric with respect to the y axis, as shown in FIG. 46. Such a deformation is called a “mode AA”, “AA” being an acronym of “asymmetric” and “asymmetric”. The combination φ_(AA) of functions that expresses this deformation is defined as follows, with an index of “AA”:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 523} \right\rbrack & \; \\ {\varphi_{AA} \equiv \left\{ \begin{matrix} \varphi_{xeo} \\ \varphi_{yoe} \end{matrix} \right.} & (1221) \end{matrix}$

11.4.21 Solution Function Using Bessel Functions of the First Kind

Eigenvalues ω₁, ω₂ are collectively represented as ω, and solutions using Bessel functions of the first kind J_(m) of eqs. (1106), (1110) are expressed as follows according to eq. (1209):

[Formula 524]

φ_(Jee)(r,θ,n,ω)≡cos 2nθ·J _(2n)(ωr)  (1222)

φ_(Joo)(r,θ,n,ω)≡sin 2nθ·J _(2n)(ωr)  (1223)

φ_(Joe)(r,θ,n,ω)≡cos(2n+1)θ·J _(2n+1)(ωr)  (1224)

φ_(Jeo)(r,θ,n,ω)≡sin(2n+1)θ·J _(2n+1)(ωr)  (1225)

Master variables of these functions are r and θ, while n and ω represent parameters. As the equation is long if all of these variables are described, only variables that should be noticed are described hereinafter. For example, in the case where n changes on both sides of an equation but r, θ, ω do not change, the argument is expressed as (n). Here, derived functions as follows are obtained:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 525} \right\rbrack & \; \\ {{\frac{\partial}{\partial x}{\varphi_{Jee}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Joe}\left( {n - 1} \right)} - {\varphi_{Joe}(n)}} \right\}}} & (1226) \\ {{\frac{\partial}{\partial x}{\varphi_{Joo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Jeo}\left( {n - 1} \right)} - {\varphi_{Jeo}(n)}} \right\}}} & (1227) \\ {{\frac{\partial}{\partial x}{\varphi_{Jeo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Joo}(n)} - {\varphi_{Joo}\left( {n + 1} \right)}} \right\}}} & (1228) \\ {{{\frac{\partial}{\partial x}{\varphi_{Joe}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Jee}(n)} - {\varphi_{Jee}\left( {n + 1} \right)}} \right\}}}{and}} & (1229) \\ {{\frac{\partial}{\partial y}{\varphi_{Jee}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Jeo}\left( {n - 1} \right)} + {\varphi_{Jeo}(n)}} \right\}}} & (1230) \\ {{\frac{\partial}{\partial y}{\varphi_{Joo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Joe}\left( {n - 1} \right)} + {\varphi_{Joe}(n)}} \right\}}} & (1231) \\ {{\frac{\partial}{\partial y}{\varphi_{Jeo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Jee}(n)} + {\varphi_{Joo}\left( {n + 1} \right)}} \right\}}} & (1232) \\ {{\frac{\partial}{\partial y}{\varphi_{Joe}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Joo}(n)} + {\varphi_{Joo}\left( {n + 1} \right)}} \right\}}} & (1233) \end{matrix}$

Further, the following relationship equations are derived from eqs. (1226) to (1229):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 526} \right\rbrack & \; \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Jee}(n)} + {\varphi_{Jee}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Joe}\left( {n - 1} \right)} - {\varphi_{Joe}\left( {n + 1} \right)}} \right\}}} & (1234) \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Joo}(n)} + {\varphi_{Joo}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Jeo}\left( {n - 1} \right)} - {\varphi_{Jeo}\left( {n + 1} \right)}} \right\}}} & (1235) \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Jeo}(n)} + {\varphi_{Jeo}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Joo}(n)} - {\varphi_{Joo}\left( {n + 2} \right)}} \right\}}} & (1236) \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Joe}(n)} + {\varphi_{Joe}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Jee}(n)} - {\varphi_{Jee}\left( {n + 2} \right)}} \right\}}} & (1237) \end{matrix}$

The following equations are derived from eqs. (1230) to (1233):

$\begin{matrix} {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Jee}(n)} - {\varphi_{Jee}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Jeo}\left( {n - 1} \right)} - {\varphi_{Jeo}\left( {n + 1} \right)}} \right\}}} & (1238) \\ \left\lbrack {{Formula}\mspace{14mu} 527} \right\rbrack & \; \\ {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Joo}(n)} - {\varphi_{Joo}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Joe}\left( {n - 1} \right)} - {\varphi_{Joe}\left( {n + 1} \right)}} \right\}}} & (1239) \\ {{\frac{\partial}{\partial y}\left\{ {{\varphi_{{Jeo}\;}(n)} - {\varphi_{Jeo}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Jee}(n)} - {\varphi_{Jee}\left( {n + 2} \right)}} \right\}}} & (1240) \\ {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Joe}(n)} - {\varphi_{Joe}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Joo}(n)} - {\varphi_{Joo}\left( {n + 2} \right)}} \right\}}} & (1241) \end{matrix}$

Here, with focus on the combinations of eqs. (1234) and (1239), eqs. (1235) and (1238), eqs. (1237) and (1240), and eqs. (1236) and (1241), the following four combinations are defined:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 528} \right\rbrack & \; \\ {{\varphi_{J\; 1{SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Jee}(n)} + {\varphi_{Jee}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Joo}(n)}} + {\varphi_{Joo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1242) \\ {{\varphi_{J\; 1{AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Joo}(n)} + {\varphi_{Joo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Jee}(n)} - {\varphi_{Jee}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1243) \\ {{\varphi_{J\; 1{SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Joe}(n)} + {\varphi_{Joe}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Jeo}(n)}} + {\varphi_{Jeo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1244) \\ {{\varphi_{J\; 1{AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Jeo}(n)} + {\varphi_{Jeo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Joe}(n)} - {\varphi_{Joe}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1245) \end{matrix}$

Then, we find that the following eq. (1120) is established for each of the combinations:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 529} \right\rbrack & \; \\ {{{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} = 0} & \begin{matrix} {\mspace{121mu} (1120)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Therefore, we can see that the combinations of eqs. (1242) to (1245) in which ω is replaced with ω₁ of eq. (1099) satisfy eqs. (1092) and (1093) of the primal simultaneous eigenvalue problem. In other words, φ_(x), φ_(y) are primal eigenfunctions. Further, according to eqs. (1151), (1152), sign-inverted functions of φ_(x), φ_(y) are dual eigenfunctions φ_(x)*, φ_(y)*. Thus, four sets of the functions φ_(x), φ_(y), φ_(x)*, φ_(y)* that satisfy the primal simultaneous differential equations (1087) and (1088) are obtained.

With use of eqs. (74) to (76), stresses can be calculated by eqs. (1242) to (1245) as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 530} \right\rbrack & \; \\ {{\sigma_{J\; 1{SA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega \left\{ {{\varphi_{Joe}\left( {n - 1} \right)} - {\varphi_{Joe}\left( {n + 1} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega \left\{ {{\varphi_{Jeo}\left( {n - 1} \right)} + {\varphi_{Jeo}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1246) \\ {{\sigma_{J\; 1{AS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega \left\{ {{\varphi_{Jeo}\left( {n - 1} \right)} - {\varphi_{Jeo}\left( {n + 1} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega \left\{ {{\varphi_{Joe}\left( {n - 1} \right)} + {\varphi_{Joe}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1247) \\ {{\sigma_{J\; 1{SS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega \left\{ {{\varphi_{Jee}(n)} - {\varphi_{Jee}\left( {n + 2} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega \left\{ {{\varphi_{Joo}(n)} + {\varphi_{Joo}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1248) \\ {{\sigma_{J\; 1{AA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega \left\{ {{\varphi_{Joo}(n)} - {\varphi_{Joo}\left( {n + 2} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Jee}(n)} + {\varphi_{Jee}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1249) \end{matrix}$

In this way, the combinations in which ω is replaced with ω₁ of eq. (1099) express stresses σ_(x), σ_(y), τ_(xy) based on the primal eigenfunctions. It should be noted that according to eqs. (82) to (84) and eqs. (1151), (1152), σ_(x), σ_(y), τ_(xy) having inverted signs represent stresses σ_(x)*, σ_(y)*, τ_(xy)* based on the dual eigenfunctions. Thus, four sets of the functions expressing stresses σ_(x), σ_(y), τ_(xy), σ_(x)*, σ_(y)*, τ_(xy)* are obtained.

The four sets are defined again below:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 531} \right\rbrack & \; \\ {{J_{1\; {SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Jee}\left( {n,\omega_{1}} \right)} + {\varphi_{Jee}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Joo}\left( {n,\omega_{1}} \right)}} + {\varphi_{Joo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Joe}\left( {{n - 1},\omega_{1}} \right)} - {\varphi_{Joe}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{\varphi_{Jeo}\left( {{n - 1},\omega_{1}} \right)} + {\varphi_{Jeo}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1250) \\ {{J_{1\; {AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Joo}\left( {n,\omega_{1}} \right)} + {\varphi_{Joo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Jee}\left( {n,\omega_{1}} \right)} - {\varphi_{Jee}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Jeo}\left( {{n - 1},\omega_{1}} \right)} - {\varphi_{Jeo}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Joe}\left( {{n - 1},\omega_{1}} \right)} + {\varphi_{Joe}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1251) \\ {{J_{1\; {SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Joe}\left( {n,\omega_{1}} \right)} + {\varphi_{Joe}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Jeo}\left( {n,\omega_{1}} \right)}} + {\varphi_{Joo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Joe}\left( {n,\omega_{1}} \right)} - {\varphi_{Jee}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{\varphi_{Joo}\left( {n,\omega_{1}} \right)} + {\varphi_{Joo}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1252) \\ {{J_{1\; {AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Jeo}\left( {n,\omega_{1}} \right)} + {\varphi_{Jeo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Joe}\left( {n,\omega_{1}} \right)} - {\varphi_{Joe}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Joo}\left( {n,\omega_{1}} \right)} - {\varphi_{Joo}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Jee}\left( {n,\omega_{1}} \right)} + {\varphi_{Jee}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1253) \end{matrix}$

Though a boundary condition has not been reflected yet, they satisfy the primal simultaneous differential equations (1087), (1088) with respect to an arbitrary n.

Similarly, focusing on combinations of eqs. (1238) and (1235), eqs. (1239) and (1234), eqs. (1241) and (1236), and eqs. (1240) and (1237), we define the following four combinations:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 532} \right\rbrack & \; \\ {{\varphi_{J\; 2{SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Jee}(n)}} + {\varphi_{Jee}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Joo}(n)} + {\varphi_{Joo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1254) \\ {{\varphi_{J\; 2{AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Joo}(n)} - {\varphi_{Joo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Jee}(n)} + {\varphi_{Jee}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1255) \\ {{\varphi_{J\; 2{SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Joe}(n)}} + {\varphi_{Joe}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Jeo}(n)} + {\varphi_{Jeo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1256) \\ {{\varphi_{J\; 2{AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Jeo}(n)} - {\varphi_{Jeo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Joe}(n)} + {\varphi_{Joe}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1257) \end{matrix}$

Regarding each of the same, the following eq. (1128) is established:

$\begin{matrix} {{{\frac{\partial}{\partial x}\varphi_{y}} - {\frac{\partial}{\partial y}\varphi_{x}}} = 0} & \begin{matrix} {\mspace{121mu} (1128\;)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Therefore, we can see that the combinations of eqs. (1254) to (1257) in which ω is replaced with ω₂ of eq. (1100) satisfy eqs. (1092) and (1093) of the primal simultaneous eigenvalue problem. In other words, φ_(x), φ_(y) are primal eigenfunctions. Further, according to eqs. (1167), (1168), sign-inverted functions of φ_(x), φ_(y) are dual eigenfunctions φ_(x)*, φ_(y)*. Thus, four sets of the functions φ_(x), φ_(y), φ_(x)*, φ_(y)* that satisfy the primal simultaneous differential equations (1087) and (1088) are obtained.

With use of eqs. (74) to (76), stresses can be calculated by eqs. (1254) to (1257) as follows:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 533} \right\rbrack} & \; \\ {{\sigma_{J\; 2{SA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{2\; \mu \; {\varphi_{Joe}(n)}} - \left\{ ~{{\varphi_{Joe}\left( {n - 1} \right)} + {\varphi_{Joe}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{2\; \mu \; {\varphi_{Joe}(n)}} + \left\{ {{\varphi_{Joe}\left( {n - 1} \right)} + {\varphi_{Joe}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Jeo}\left( {n - 1} \right)} - {\varphi_{Jeo}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1258) \\ {{\sigma_{J\; 2{AS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{{- 2}\; \mu \; {\varphi_{Jeo}(n)}} + \left\{ ~{{\varphi_{Jeo}\left( {n - 1} \right)} + {\varphi_{Jeo}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{{- 2}\; \mu \; {\varphi_{Jeo}(n)}} - \left\{ {{\varphi_{Jeo}\left( {n - 1} \right)} + {\varphi_{Jeo}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Joe}\left( {n - 1} \right)} - {\varphi_{Joe}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1259) \\ {{\sigma_{J\; 2{SS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{2\; \mu \; {\varphi_{Jee}\left( {n + 1} \right)}} - \left\{ ~{{\varphi_{Jee}(n)} + {\varphi_{Jee}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{2\; \mu \; {\varphi_{Jee}\left( {n + 1} \right)}} + \left\{ {{\varphi_{Jee}(n)} + {\varphi_{Jee}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Joo}(n)} - {\varphi_{Joo}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1260) \\ {{\sigma_{J\; 2{AA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{{- 2}\; \mu \; {\varphi_{Joo}\left( {n + 1} \right)}} + \left\{ ~{{\varphi_{Joo}(n)} + {\varphi_{Joo}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{{- 2}\; \mu \; {\varphi_{Joo}\left( {n + 1} \right)}} - \left\{ {{\varphi_{Joo}(n)} + {\varphi_{Joo}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Jee}(n)} - {\varphi_{Jee}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1261) \end{matrix}$

In this way, the combinations in which ω is replaced with ω₂ of eq. (1100) express stresses σ_(x), σ_(y), τ_(xy) based on the primal eigenfunctions. It should be noted that according to eqs. (82) to (84) and eqs. (1167), (1168), σ_(x), σ_(y), τ_(xy) having inverted signs represent stresses σ_(x)*, σ_(y)*, τ_(xy)* based on the dual eigenfunctions. Thus, four sets of the functions expressing stresses σ_(x), σ_(y), τ_(xy), σ_(x)*, σ_(y)*, τ_(xy)* are obtained.

The four sets are defined again below:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 534} \right\rbrack} & \; \\ {{J_{2{SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Jee}\left( {n,\omega_{2}} \right)}} + {\varphi_{Jee}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Joo}\left( {n,\omega_{2}} \right)} + {\varphi_{Joo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{2\; \mu \; {\varphi_{Joe}\left( {n,\omega_{2}} \right)}} - \left\{ {{\varphi_{Joe}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Joe}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{2\; \mu \; {\varphi_{Joe}\left( {n,\omega_{2}} \right)}} + \left\{ {{\varphi_{Joe}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Joe}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Jeo}\left( {{n - 1},\omega_{2}} \right)} - {\varphi_{Jeo}\left( {{n + 1},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1262) \\ {{J_{2{AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Joo}\left( {n,\omega_{2}} \right)} - {\varphi_{Joo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Jee}\left( {n,\omega_{2}} \right)} + {\varphi_{Jee}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}\; \mu \; {\varphi_{Jeo}\left( {n,\omega_{2}} \right)}} + \left\{ {{\varphi_{Jeo}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Jeo}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}\; \mu \; {\varphi_{Jeo}\left( {n,\omega_{2}} \right)}} - \left\{ {{\varphi_{Jeo}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Jeo}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Joe}\left( {{n - 1},\omega_{2}} \right)} - {\varphi_{Joe}\left( {{n + 1},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1263) \\ {{J_{2{SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Joe}\left( {n,\omega_{2}} \right)}} + {\varphi_{Joe}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Jeo}\left( {n,\omega_{2}} \right)} + {\varphi_{Jeo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{2\; \mu \; {\varphi_{Jee}\left( {{n + 1},\omega_{2}} \right)}} - \left\{ {{\varphi_{Jee}\left( {n,\omega_{2}} \right)} + {\varphi_{Jee}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{2\; \mu \; {\varphi_{Jee}\left( {{n + 1},\omega_{2}} \right)}} + \left\{ {{\varphi_{Jee}\left( {n,\omega_{2}} \right)} + {\varphi_{Jee}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Joo}\left( {n,\omega_{2}} \right)} - {\varphi_{Joo}\left( {{n + 2},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1264) \\ {{J_{2{AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Jeo}\left( {n,\omega_{2}} \right)} - {\varphi_{Jeo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Joe}\left( {n,\omega_{2}} \right)} + {\varphi_{Joe}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}\; \mu \; {\varphi_{Joo}\left( {{n + 1},\omega_{2}} \right)}} + \left\{ {{\varphi_{Joo}\left( {n,\omega_{2}} \right)} + {\varphi_{Joo}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}\; \mu \; {\varphi_{Joo}\left( {{n + 1},\omega_{2}} \right)}} - \left\{ {{\varphi_{Joo}\left( {n,\omega_{2}} \right)} + {\varphi_{Joo}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Jee}\left( {n,\omega_{2}} \right)} - {\varphi_{Jee}\left( {{n + 2},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1265) \end{matrix}$

Though a boundary condition has not been reflected yet, they satisfy the primal simultaneous differential equations (1087), (1088) with respect to an arbitrary n.

11.4.22 Solution Function Using Bessel Functions of the Second Kind

Eigenvalues ω₁, ω₂ are collectively represented as w, and solutions using Bessel functions of the second kind Y_(m) of eqs. (1106), (1110) are expressed as follows according to eq. (1209):

[Formula 535]

φ_(Yee)(r,θ,n,ω)≡cos 2n θ·Y _(2n)(ωr)  (1266)

[Formula 536]

φ_(Yoo)(r,θ,n,ω)≡sin 2nθ·Y _(2n)(ωr)  (1267)

φ_(Yoe)(r,θ,n,ω)≡cos(2n+1)θ·Y _(2n+1)(ωr)  (1268)

φ_(Yeo)(r,θ,n,ω)≡sin(2n+1)θ·Y _(2n+1)(ωr)  (1269)

Master variables of these functions are r and θ, while n and ω represent parameters. As the equation is long if all of these variables are described, only variables that should be noticed are described hereinafter. For example, in the case where n changes on both sides of an equation but r, θ, ω do not change, the argument is expressed as (n). Here, derived functions as follows are obtained:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 537} \right\rbrack & \; \\ {{\frac{\partial}{\partial x}{\varphi_{Yee}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yoe}\left( {n - 1} \right)} - {\varphi_{Yoe}(n)}} \right\}}} & (1270) \\ {{\frac{\partial}{\partial x}{\varphi_{Yoo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yeo}\left( {n - 1} \right)} - {\varphi_{Yeo}(n)}} \right\}}} & (1271) \\ {{\frac{\partial}{\partial x}{\varphi_{Yeo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yoo}(n)} - {\varphi_{Yoo}\left( {n + 1} \right)}} \right\}}} & (1272) \\ {{\frac{\partial}{\partial x}{\varphi_{Yoe}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yee}(n)} - {\varphi_{Yee}\left( {n + 1} \right)}} \right\}}} & (1273) \\ {and} & \; \\ {{\frac{\partial}{\partial y}{\varphi_{Yee}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Yeo}\left( {n - 1} \right)} + {\varphi_{Yeo}(n)}} \right\}}} & (1274) \\ {{\frac{\partial}{\partial y}{\varphi_{Yoo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yoe}\left( {n - 1} \right)} + {\varphi_{Yoe}(n)}} \right\}}} & (1275) \\ {{\frac{\partial}{\partial y}{\varphi_{Yeo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yee}(n)} + {\varphi_{Yee}\left( {n + 1} \right)}} \right\}}} & (1276) \\ {{\frac{\partial}{\partial y}{\varphi_{Yoe}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Yoo}(n)} + {\varphi_{Yoo}\left( {n + 1} \right)}} \right\}}} & (1277) \end{matrix}$

Further, the following relationship equations are derived from eqs. (1270) to (1273):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 538} \right\rbrack & \; \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Yee}(n)} + {\varphi_{Yee}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yoe}\left( {n - 1} \right)} - {\varphi_{Yoe}\left( {n + 1} \right)}} \right\}}} & (1278) \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Yoo}(n)} + {\varphi_{Yoo}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yeo}\left( {n - 1} \right)} - {\varphi_{Yeo}\left( {n + 1} \right)}} \right\}}} & (1279) \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Yeo}(n)} + {\varphi_{Yeo}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yoo}(n)} - {\varphi_{Yoo}\left( {n + 2} \right)}} \right\}}} & (1280) \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Yoe}(n)} + {\varphi_{Yoe}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yee}(n)} - {\varphi_{Yee}\left( {n + 2} \right)}} \right\}}} & (1281) \end{matrix}$

The following equations are derived from eqs. (1274) to (1277):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 539} \right\rbrack & \; \\ {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Yee}(n)} - {\varphi_{Yee}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Yeo}\left( {n - 1} \right)} - {\varphi_{Yeo}\left( {n + 1} \right)}} \right\}}} & (1282) \\ {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Yoo}(n)} - {\varphi_{Yoo}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yoe}\left( {n - 1} \right)} - {\varphi_{Yoe}\left( {n + 1} \right)}} \right\}}} & (1283) \\ {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Yeo}(n)} - {\varphi_{Yeo}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Yee}(n)} - {\varphi_{Yee}\left( {n + 2} \right)}} \right\}}} & (1284) \\ {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Yoe}(n)} - {\varphi_{Yoe}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Yoo}(n)} - {\varphi_{Yoo}\left( {n + 2} \right)}} \right\}}} & (1285) \end{matrix}$

Here, with focus on the combinations of eqs. (1278) and (1283), eqs. (1279) and (1282), eqs. (1281) and (1284), and eqs. (1280) and (1285), the following four combinations are defined:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 540} \right\rbrack & \; \\ {{\varphi_{Y\; 1{SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yee}(n)} + {\varphi_{Yee}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Yoo}(n)}} + {\varphi_{Yoo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1286) \\ {{\varphi_{Y\; 1{AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yoo}(n)} + {\varphi_{Yoo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yee}(n)} - {\varphi_{Yee}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1287) \\ {{\varphi_{Y\; 1{SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yoe}(n)} + {\varphi_{Yoe}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Yeo}(n)}} + {\varphi_{Yeo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1288) \\ {{\varphi_{Y\; 1{AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yeo}(n)} + {\varphi_{Yeo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yoe}(n)} - {\varphi_{Yoe}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1289) \end{matrix}$

Then, we find that the following eq. (1120) is established for each of the combinations:

$\begin{matrix} {{{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} = 0} & \begin{matrix} {\mspace{115mu} (1120)} \\ ({Aforementioned}) \end{matrix} \end{matrix}$

Therefore, we can see that the combinations of eqs. (1286) to (1289) in which ω is replaced with ω₁ of eq. (1099) satisfy eqs. (1092) and (1093) of the primal simultaneous eigenvalue problem. In other words, φ_(x), φ_(y) are primal eigenfunctions. Further, according to eqs. (1151), (1152), sign-inverted functions of φ_(x), φ_(y) are dual eigenfunctions φ_(x)*, φ_(y)*. Thus, four sets of the functions φ_(x), φ_(y), φ_(x)*, φ_(y)* that satisfy the primal simultaneous differential equations (1087) and (1088) are obtained.

With use of eqs. (74) to (76), stresses can be calculated by eqs. (1286) to (1289) as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 541} \right\rbrack & \; \\ {{\sigma_{Y\; 1{SA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega \left\{ {{\varphi_{Yoe}\left( {n - 1} \right)} - {\varphi_{Yoe}\left( {n + 1} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega \left\{ {{\varphi_{Yeo}\left( {n - 1} \right)} + {\varphi_{Yeo}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1290) \\ {{\sigma_{Y\; 1{AS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega \left\{ {{\varphi_{Yeo}\left( {n - 1} \right)} - {\varphi_{Yeo}\left( {n + 1} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Yoe}\left( {n - 1} \right)} + {\varphi_{Yoe}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1291) \\ {{\sigma_{Y\; 1{SS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega \left\{ {{\varphi_{Yee}(n)} - {\varphi_{Yee}\left( {n + 2} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega \left\{ {{\varphi_{Yoo}(n)} + {\varphi_{Yoo}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1292) \\ {{\sigma_{Y\; 1{AA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega \left\{ {{\varphi_{Yoo}(n)} - {\varphi_{Yoo}\left( {n + 2} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Yee}(n)} + {\varphi_{Yee}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1293) \end{matrix}$

In this way, the combinations in which ω is replaced with ω₁ of eq. (1099) express stresses σ_(x), σ_(y), τ_(xy) based on the primal eigenfunctions. It should be noted that according to eqs. (82) to (84) and eqs. (1151), (1152), σ_(x), σ_(y), τ_(xy) having inverted signs represent stresses σ_(x)*, σ_(y)*, τ_(xy) ^(*) based on the dual eigenfunctions. Thus, four sets of the functions expressing stresses σ_(x), σ_(y), τ_(xy), σ_(x)*, σ_(y)*, τ_(xy)* are obtained.

The four sets are defined again below:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 542} \right\rbrack & \; \\ {{Y_{1\; {SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yee}\left( {n,\omega_{1}} \right)} + {\varphi_{Yee}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Yoo}\left( {n,\omega_{1}} \right)}} + {\varphi_{Yoo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Yoe}\left( {{n - 1},\omega_{1}} \right)} - {\varphi_{Yoe}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{\varphi_{Yeo}\left( {{n - 1},\omega_{1}} \right)} + {\varphi_{Yeo}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1294) \\ {{Y_{1\; {AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yoo}\left( {n,\omega_{1}} \right)} + {\varphi_{Yoo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yee}\left( {n,\omega_{1}} \right)} - {\varphi_{Yee}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Yeo}\left( {{n - 1},\omega_{1}} \right)} - {\varphi_{Yeo}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Yoe}\left( {{n - 1},\omega_{1}} \right)} + {\varphi_{Yoe}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1295) \\ {{Y_{1\; {SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yoe}\left( {n,\omega_{1}} \right)} + {\varphi_{Yoe}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Yeo}\left( {n,\omega_{1}} \right)}} + {\varphi_{Yeo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Yee}\left( {n,\omega_{1}} \right)} - {\varphi_{Yee}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{\varphi_{Yoo}\left( {n,\omega_{1}} \right)} + {\varphi_{Yoo}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1296) \\ \left\lbrack {{Formula}\mspace{14mu} 543} \right\rbrack & \; \\ {{Y_{1\; {AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yeo}\left( {n,\omega_{1}} \right)} + {\varphi_{Yeo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yoe}\left( {n,\omega_{1}} \right)} - {\varphi_{Yoe}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Yoo}\left( {n,\omega_{1}} \right)} - {\varphi_{Yoo}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Yee}\left( {n,\omega_{1}} \right)} + {\varphi_{Yee}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1297) \end{matrix}$

Though a boundary condition has not been reflected yet, they satisfy the primal simultaneous differential equations (1087), (1088) with respect to an arbitrary n.

Similarly, focusing on combinations of eqs. (1282) and (1279), eqs. (1283) and (1278), eqs. (1285) and (1280), and eqs. (1284) and (1281), we define the following four combinations:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 544} \right\rbrack & \; \\ {{\varphi_{Y\; 2{SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Yee}(n)}} + {\varphi_{Yee}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yoo}(n)} + {\varphi_{Yoo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1298) \\ {{\varphi_{Y\; 2{AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yoo}(n)} - {\varphi_{Yoo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yee}(n)} + {\varphi_{Yee}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1299) \\ {{\varphi_{Y\; 2{SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Yoe}(n)}} + {\varphi_{Yoe}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yeo}(n)} + {\varphi_{Yeo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1300) \\ {{\varphi_{Y\; 2{AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yeo}(n)} - {\varphi_{Yeo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yoe}(n)} + {\varphi_{Yoe}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1301) \end{matrix}$

Regarding each of the same, the following eq. (1128) is established:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 544} \right\rbrack & \; \\ {{\varphi_{Y\; 2{SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Yee}(n)}} + {\varphi_{Yee}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yoo}(n)} + {\varphi_{Yoo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1298) \\ {{\varphi_{Y\; 2{AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yoo}(n)} - {\varphi_{Yoo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yee}(n)} + {\varphi_{Yee}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1299) \\ {{\varphi_{Y\; 2{SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Yoe}(n)}} + {\varphi_{Yoe}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yeo}(n)} + {\varphi_{Yeo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1300) \\ {{\varphi_{Y\; 2{AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yeo}(n)} - {\varphi_{Yeo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{Y_{Yoe}(n)} + {\varphi_{Yoe}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1301) \end{matrix}$

Therefore, we can see that the combinations of eqs. (1298) to (1301) in which ω is replaced with ω₂ of eq. (1100) satisfy the primal simultaneous eigenvalue problem of eqs. (1092) and (1093). In other words, φ_(x), φ_(y) are primal eigenfunctions. Further, according to eqs. (1167), (1168), sign-inverted functions of φx, φ_(y) are dual eigenfunctions φ_(x)*, φ_(y)*. Thus, four sets of the functions φ_(x), φ_(y), φ_(x)*, φ_(y)* that satisfy the primal simultaneous differential equations (1087) and (1088) are obtained.

Further, with use of eqs. (74) to (76), stresses can be calculated by eqs. (1298) to (1301) as follows:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 545} \right\rbrack} & \; \\ {{\sigma_{Y\; 2{SA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{2{{\mu\varphi}_{Yoe}(n)}} - \left\{ {{\varphi_{Yoe}\left( {n - 1} \right)} + {\varphi_{Yoe}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{2{{\mu\varphi}_{Yoe}(n)}} + \left\{ {{\varphi_{Yoe}\left( {n - 1} \right)} + {\varphi_{Yoe}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Yeo}\left( {n - 1} \right)} - {\varphi_{Yeo}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1302) \\ {{\sigma_{Y\; 2{AS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{{- 2}\mu \; {\varphi_{Yeo}(n)}} + \left\{ {{\varphi_{Yeo}\left( {n - 1} \right)} + {\varphi_{Yeo}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{{- 2}\mu \; {\varphi_{Yeo}(n)}} - \left\{ {{\varphi_{Yeo}\left( {n - 1} \right)} + {\varphi_{Yeo}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Yoe}\left( {n - 1} \right)} - {\varphi_{Yoe}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1303) \\ {{\sigma_{Y\; 2{SS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{2{{\mu\varphi}_{Yee}\left( {n + 1} \right)}} - \left\{ {{\varphi_{Yee}(n)} + {\varphi_{Yee}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{2{{\mu\varphi}_{Yee}\left( {n + 1} \right)}} + \left\{ {{\varphi_{Yee}(n)} + {\varphi_{Yee}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Yoo}(n)} - {\varphi_{Yoo}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1304) \\ {{\sigma_{Y\; 2{AA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{{- 2}{{\mu\varphi}_{Yoo}\left( {n + 1} \right)}} + \left\{ {{\varphi_{Yoo}(n)} + {\varphi_{Yoo}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{{- 2}\mu \; {\varphi_{Yoo}\left( {n + 1} \right)}} - \left\{ {{\varphi_{Yoo}(n)} + {\varphi_{Yoo}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Yee}(n)} - {\varphi_{Yee}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1305) \end{matrix}$

In this way, the combinations in which ω is replaced with ω₂ of eq. (1100) express stresses σ_(x), σ_(y), τ_(xy) based on the primal eigenfunctions. It should be noted that according to eqs. (82) to (84) and eqs. (1167), (1168), σ_(x), σ_(y), τ_(xy) having inverted signs represent stresses σ_(x)*, σ_(y)*, τ_(xy)* based on the dual eigenfunctions. Thus, four sets of the functions expressing stresses σ_(x), σ_(y), τ_(xy), σ_(x)*, σ_(y)*, τ_(xy)* are obtained.

The four sets are defined again below:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 546} \right\rbrack} & \; \\ {{Y_{2{SA}}(n)} = \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Yee}\left( {n,\omega_{2}} \right)}} + {\varphi_{Yee}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yoo}\left( {n,\omega_{2}} \right)} + {\varphi_{Yoo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Yoe}\left( {n,\omega_{2}} \right)}} - \left\{ {{\varphi_{Yoe}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Yoe}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Yoe}\left( {n,\omega_{2}} \right)}} + \left\{ {{\varphi_{Yoe}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Yoe}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Yeo}\left( {{n - 1},\omega_{2}} \right)} - {\varphi_{Yeo}\left( {{n + 1},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \varphi_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1306) \\ {{Y_{2{AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yoo}\left( {n,\omega_{2}} \right)} - {\varphi_{Yoo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yee}\left( {n,\omega_{2}} \right)} + {\varphi_{Yee}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}\mu \; {\varphi_{Yeo}\left( {n,\omega_{2}} \right)}} + \left\{ {{\varphi_{Yeo}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Yeo}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}{{\mu\varphi}_{Yeo}\left( {n,\omega_{2}} \right)}} - \left\{ {{\varphi_{Yeo}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Yeo}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Yoe}\left( {{n - 1},\omega_{2}} \right)} - {\varphi_{Yoe}\left( {{n + 1},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1307) \\ {{Y_{2{SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Yoe}\left( {n,\omega_{2}} \right)}} + {\varphi_{Yoe}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yeo}\left( {n,\omega_{2}} \right)} + {\varphi_{Yeo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Yee}\left( {{n + 1},\omega_{2}} \right)}} - \left\{ {{\varphi_{Yee}\left( {n,\omega_{2}} \right)} + {\varphi_{Yee}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Yee}\left( {{n + 1},\omega_{2}} \right)}} + \left\{ {{\varphi_{Yee}\left( {n,\omega_{2}} \right)} + {\varphi_{Yee}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Y_{oo}}\left( {n,\omega_{2}} \right)} - {\varphi_{Yoo}\left( {{n + 2},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1308) \\ {{Y_{2{AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Yeo}\left( {n,\omega_{2}} \right)} - {\varphi_{Yeo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Yoe}\left( {n,\omega_{2}} \right)} + {\varphi_{Yoe}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}\mu \; {\varphi_{Yoo}\left( {{n + 1},\omega_{2}} \right)}} + \left\{ {{\varphi_{Yoo}\left( {n,\omega_{2}} \right)} + {\varphi_{Yoo}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}\mu \; {\varphi_{Yoo}\left( {{n + 1},\omega_{2}} \right)}} - \left\{ {{\varphi_{Yoo}\left( {n,\omega_{2}} \right)} + {\varphi_{Yoo}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Yee}\left( {n,\omega_{2}} \right)} - {\varphi_{Yee}\left( {{n + 2},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right.} & (1309) \end{matrix}$

Though a boundary condition has not been reflected yet, they satisfy the primal simultaneous differential equations (1087), (1088) with respect to an arbitrary n.

11.4.23 Solution Function Using Modified Bessel Function of the First Kind

Eigenvalues ω₁, ω₂ are collectively represented as ω, and solutions using the modified Bessel function of the first kind I_(m) of eqs. (1108), (1112) are expressed as follows according to eq. (1209):

[Formula 547]

φ_(Iee)(r,θ,n,ω)≡cos 2nθ·I _(2n)(ωr)  (1310)

φ_(Ioo)(r,θ,n,ω)≡sin 2nθ·I _(2n)(ωr)  (1311)

φ_(Ioe)(r,θ,n,ω)≡cos(2n+1)θ·I _(2n+1)(ωr)  (1312)

φ_(Ieo)(r,θ,n,ω)≡sin(2n+1)θ·I _(2n+1)(ωr)  (1313)

Master variables of these functions are r and θ, while n and ω represent parameters. As the equation is long if all of these variables are described, only variables that should be noticed are described hereinafter. For example, in the case where n changes on both sides of an equation but r, θ, ω do not change, the argument is expressed as (n). Here, derived functions as follows are obtained:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 548} \right\rbrack & \; \\ {{\frac{\partial}{\partial x}{\varphi_{Iee}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Ioe}\left( {n - 1} \right)} + {\varphi_{Ioe}(n)}} \right\}}} & (1314) \\ {{\frac{\partial}{\partial x}{\varphi_{Ioo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Ieo}\left( {n - 1} \right)} + {\varphi_{Ieo}(n)}} \right\}}} & (1315) \\ {{\frac{\partial}{\partial x}{\varphi_{leo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{loo}(n)} + {\varphi_{loo}\left( {n + 1} \right)}} \right\}}} & (1316) \\ {{{\frac{\partial}{\partial x}{\varphi_{Ioe}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Iee}(n)} + {\varphi_{Iee}\left( {n + 1} \right)}} \right\}}}{and}} & (1317) \\ {{\frac{\partial}{\partial y}{\varphi_{Iee}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Ieo}\left( {n - 1} \right)} - {\varphi_{Ieo}(n)}} \right\}}} & (1318) \\ {{\frac{\partial}{\partial y}{\varphi_{Ioo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Ioe}\left( {n - 1} \right)} - {\varphi_{Ioe}(n)}} \right\}}} & (1319) \\ {{\frac{\partial}{\partial y}{\varphi_{Ieo}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Iee}(n)} - {\varphi_{Iee}\left( {n + 1} \right)}} \right\}}} & (1320) \\ \left\lbrack {{Formula}\mspace{14mu} 549} \right\rbrack & \; \\ {{\frac{\partial}{\partial y}{\varphi_{Ioe}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Ioo}(n)} - {\varphi_{Ioo}\left( {n + 1} \right)}} \right\}}} & (1321) \end{matrix}$

Further, the following relationship equations are derived from eqs. (1314) to (1317):

$\begin{matrix} {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Iee}(n)} - {\varphi_{Iee}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Ioe}\left( {n - 1} \right)} - {\varphi_{Ioe}\left( {n + 1} \right)}} \right\}}} & (1322) \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Ioo}(n)} - {\varphi_{Ioo}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Ieo}\left( {n - 1} \right)} - {\varphi_{Ieo}\left( {n + 1} \right)}} \right\}}} & (1323) \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Ieo}(n)} - {\varphi_{Ieo}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Ioo}(n)} - {\varphi_{Ioo}\left( {n + 2} \right)}} \right\}}} & (1324) \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Ioe}(n)} - {\varphi_{Ioe}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Iee}(n)} - {\varphi_{Iee}\left( {n + 2} \right)}} \right\}}} & (1325) \end{matrix}$

The following relationship equations are derived from eqs. (1318) to (1321):

$\begin{matrix} {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Iee}(n)} + {\varphi_{Iee}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Ieo}\left( {n - 1} \right)} - {\varphi_{Ieo}\left( {n + 1} \right)}} \right\}}} & (1326) \\ {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Ioo}(n)} + {\varphi_{Ioo}\left( {n + 1} \right)}} \right)} = {\frac{\omega}{2}\left\{ {{\varphi_{Ioe}\left( {n - 1} \right)} - {\varphi_{Ioe}\left( {n + 1} \right)}} \right\}}} & (1327) \\ {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Ieo}(n)} + {\varphi_{Ieo}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Iee}(n)} - {\varphi_{Iee}\left( {n + 2} \right)}} \right\}}} & (1328) \\ {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Ioe}(n)} + {\varphi_{Ioe}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Ioo}(n)} - {\varphi_{Ioo}\left( {n + 2} \right)}} \right\}}} & (1329) \end{matrix}$

Here, with focus on the combinations of eqs. (1322) and (1327), eqs. (1323) and (1326), eqs. (1325) and (1328), and eqs. (1324) and (1329), the following four combinations are defined:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 550} \right\rbrack & \; \\ {{\varphi_{I\; 1{SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Iee}(n)}} + {\varphi_{Iee}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Ioo}(n)} + {\varphi_{Ioo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1330) \\ {{\varphi_{I\; 1{AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Ioo}(n)} - {\varphi_{Ioo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Iee}(n)} + {\varphi_{Iee}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1331) \\ {{\varphi_{I\; 1{SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Ioe}(n)}} + {\varphi_{Ioe}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Ieo}(n)} + {\varphi_{Ieo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1332) \\ {{\varphi_{I\; 1{AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Ieo}(n)} - {\varphi_{Ieo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Ioe}(n)} + {\varphi_{Ioe}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1333) \end{matrix}$

Then, we find that the following eq. (1120) is established for each of the combinations:

$\begin{matrix} {{{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} = 0} & \underset{({Aforementioned})}{(1120)} \end{matrix}$

Therefore, we can see that the combinations of eqs. (1330) to (1333) in which ω is replaced with ω₁ of eq. (1099) satisfy eqs. (1092) and (1093) of the primal simultaneous eigenvalue problem. In other words, φ_(x), φ_(y) are primal eigenfunctions. Further, according to eqs. (1157), (1158), the same functions as φ_(x), φ_(y), are dual eigenfunctions φ_(x)*, φ_(y)*. Thus, four sets of the functions φ_(x), φ_(y), φ_(x)*, φ_(y)* that satisfy the primal simultaneous differential equations (1087) and (1088) are obtained.

With use of eqs. (74) to (76), stresses can be calculated by eq. (1330) to eq. (1333) as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 551} \right\rbrack & \; \\ {{\sigma_{I\; 1{SA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega \left\{ {{- {\varphi_{Ioe}\left( {n - 1} \right)}} + {\varphi_{Ioe}\left( {n + 1} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Ieo}\left( {n - 1} \right)} + {\varphi_{Ieo}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1334) \\ {{\sigma_{I\; 1{AS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega \left\{ {{\varphi_{Ieo}\left( {n - 1} \right)} - {\varphi_{Ieo}\left( {n + 1} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Ioe}\left( {n - 1} \right)} + {\varphi_{Ioe}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1335) \\ {{\sigma_{I\; 1{SS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {{- G}\; \omega \left\{ {{\varphi_{Iee}(n)} - {\varphi_{Iee}\left( {n + 2} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Ioo}(n)} + {\varphi_{Ioo}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1336) \\ {{\sigma_{I\; 1{AA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega \left\{ {{\varphi_{Ioo}(n)} - {\varphi_{Ioo}\left( {n + 2} \right)}} \right\}}} \\ {\sigma_{y} = {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Iee}(n)} + {\varphi_{Iee}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1337) \end{matrix}$

In this way, the combinations in which ω is replaced with ω₁ of eq. (1099) express stresses σ_(x), σ_(y), τ_(xy) based on the primal eigenfunctions. It should be noted that according to eqs. (82) to (84) and eqs. (1157), (1158), the same as σ_(x), σ_(y), τ_(xy) represent stresses σ_(x)*, σ_(y)*, τ_(xy)* based on the dual eigenfunctions. Thus, four sets of the functions expressing stresses σ_(x), σ_(y), τ_(xy), σ_(x)*, σ_(y)*, τ_(xy)* are obtained.

The four sets are defined again below:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 552} \right\rbrack & \; \\ {{I_{1{SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Iee}\left( {n,\omega_{1}} \right)}} + {\varphi_{Iee}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Ioo}\left( {n,\omega_{1}} \right)} + {\varphi_{Ioo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{- {\varphi_{Ioe}\left( {{n - 1},\omega_{1}} \right)}} + {\varphi_{Ioe}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Ieo}\left( {{n - 1},\omega_{1}} \right)} + {\varphi_{Ieo}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1338) \\ {{I_{1{AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Ioo}\left( {n,\omega_{1}} \right)} - {\varphi_{Ioo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Iee}\left( {n,\omega_{1}} \right)} + {\varphi_{Iee}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Ieo}\left( {{n - 1},\omega_{1}} \right)} - {\varphi_{Ieo}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Ioe}\left( {{n - 1},\omega_{1}} \right)} + {\varphi_{Ioe}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1339) \\ {{I_{1{SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Ioe}\left( {n,\omega_{1}} \right)}} + {\varphi_{Ioe}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Ieo}\left( {n,\omega_{1}} \right)} + {\varphi_{Ieo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {{- G}\; \omega_{1}\left\{ {{\varphi_{Iee}\left( {n,\omega_{1}} \right)} - {\varphi_{Iee}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Ioo}\left( {n,\omega_{1}} \right)} + {\varphi_{Ioo}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1340) \\ {{I_{1{AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Ieo}\left( {n,\omega_{1}} \right)} - {\varphi_{Ieo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Ioe}\left( {n,\omega_{1}} \right)} + {\varphi_{Ioe}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Ioo}\left( {n,\omega_{1}} \right)} - {\varphi_{Ioo}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Iee}\left( {n,\omega_{1}} \right)} + {\varphi_{Iee}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} = \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1341) \end{matrix}$

Though a boundary condition has not been reflected yet, they satisfy the primal simultaneous differential equations (1087), (1088) with respect to an arbitrary n.

Similarly, focusing on combinations of eqs. (1326) and (1323), eqs. (1327) and (1322), eqs. (1329) and (1324), and eqs. (1328) and (1325), we define the following four combinations:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 553} \right\rbrack & \; \\ {{\varphi_{I\; 2{SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Iee}(n)} + {\varphi_{Iee}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Ioo}(n)}} + {\varphi_{Ioo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1342) \\ {{\varphi_{I\; 2{AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Ioo}(n)} + {\varphi_{Ioo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Iee}(n)} - {\varphi_{Iee}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1343) \\ {{\varphi_{I\; 2{SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Ioe}(n)} + {\varphi_{Ioe}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Ieo}(n)}} + {\varphi_{Ieo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1344) \\ {{\varphi_{I\; 2{AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Ieo}(n)} + {\varphi_{Ieo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Ioe}(n)} - {\varphi_{Ioe}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1345) \end{matrix}$

Regarding each of the same, the following eq. (1128) is established:

$\begin{matrix} {{{\frac{\partial}{\partial x}\varphi_{y}} - {\frac{\partial}{\partial y}\varphi_{x}}} = 0} & \underset{({Aforementioned})}{(1128)} \end{matrix}$

Therefore, we can see that the combinations of eqs. (1342) to (1345) in which ω is replaced with ω₂ of eq. (1100) satisfy eqs. (1092) and (1093) of the primal simultaneous eigenvalue problem. In other words, φ_(x), φ_(y) are primal eigenfunctions. Further, according to eqs. (1173), (1174), the same functions as φ_(x), φ_(y) are dual eigenfunctions φ_(x)*, φ_(y)*. Thus, four sets of the functions φ_(x), φ_(y), φ_(x)*, φ_(y)* that satisfy the primal simultaneous differential equations (1087) and (1088) are obtained.

With use of eqs. (74) to (76), stresses can be calculated by eqs. (1342) to (1345) as follows:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 554} \right\rbrack} & \; \\ {{\sigma_{I\; 2{SA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{2{{\mu\varphi}_{Ioe}(n)}} + \left\{ {{\varphi_{Ioe}\left( {n - 1} \right)} + {\varphi_{Ioe}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{2{{\mu\varphi}_{Ioe}(n)}} - \left\{ {{\varphi_{Ioe}\left( {n - 1} \right)} + {\varphi_{Ioe}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{- {\varphi_{Ieo}\left( {n - 1} \right)}} + {\varphi_{Ieo}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1346) \\ {{\sigma_{I\; 2{AS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{2{{\mu\varphi}_{Ieo}(n)}} + \left\{ {{\varphi_{Ieo}\left( {n - 1} \right)} + {\varphi_{Ieo}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{2\mu \; {\varphi_{Ieo}(n)}} - \left\{ {{\varphi_{Ieo}\left( {n - 1} \right)} + {\varphi_{Ieo}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Ioe}\left( {n - 1} \right)} - {\varphi_{Ioe}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1347) \\ {{\sigma_{I\; 2{SS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{2{{\mu\varphi}_{Iee}\left( {n + 1} \right)}} + \left\{ {{\varphi_{Iee}(n)} + {\varphi_{Iee}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{2{{\mu\varphi}_{Iee}\left( {n + 1} \right)}} - \left\{ {{\varphi_{Iee}(n)} + {\varphi_{Iee}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{- {\varphi_{Ioo}(n)}} + {\varphi_{Ioo}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1348) \\ {{\sigma_{I\; 2{AA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{2{{\mu\varphi}_{Ioo}\left( {n + 1} \right)}} + \left\{ {{\varphi_{Ioo}(n)} + {\varphi_{Ioo}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{2{{\mu\varphi}_{Ioo}\left( {n + 1} \right)}} - \left\{ {{\varphi_{Ioo}(n)} + {\varphi_{Ioo}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Iee}(n)} - {\varphi_{Iee}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1349) \end{matrix}$

In this way, the combinations in which ω is replaced with ω₂ of eq. (1100) express stresses σ_(x), σ_(y), τ_(xy) based on the primal eigenfunctions. It should be noted that according to eqs. (82) to (84) and eqs. (1173), (1174), the same as σ_(x), σ_(y), τ_(xy) represent stresses σ_(x)*, σ_(y)*, τ_(xy)* based on the dual eigenfunctions. Thus, four sets of the functions expressing stresses σ_(x), σ_(y), τ_(xy), σ_(x)*, σ_(y)*, τ_(xy)* are obtained.

The four sets are defined again below:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 555} \right\rbrack} & \; \\ {{I_{2{SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Iee}\left( {n,\omega_{2}} \right)} + {\varphi_{Iee}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Ioo}\left( {n,\omega_{2}} \right)}} + {\varphi_{Ioo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Ioe}\left( {n,\omega_{2}} \right)}} + \left\{ {{\varphi_{Ioe}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Ioe}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Ioe}\left( {n,\omega_{2}} \right)}} - \left\{ {{\varphi_{Ioe}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Ioe}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{- {\varphi_{Ieo}\left( {{n - 1},\omega_{2}} \right)}} + {\varphi_{Ieo}\left( {{n + 1},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1350) \\ {{I_{2{AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Ioo}\left( {n,\omega_{2}} \right)} + {\varphi_{Ioo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Iee}\left( {n,\omega_{2}} \right)} - {\varphi_{Iee}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Ieo}\left( {n,\omega_{2}} \right)}} + \left\{ {{\varphi_{Ieo}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Ieo}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Ieo}\left( {n,\omega_{2}} \right)}} - \left\{ {{\varphi_{Ieo}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Ieo}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Ioe}\left( {{n - 1},\omega_{2}} \right)} - {\varphi_{Ioe}\left( {{n + 1},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1351) \\ {{I_{2{SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Ioe}\left( {n,\omega_{2}} \right)} + {\varphi_{Ioe}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Ieo}\left( {n,\omega_{2}} \right)}} + {\varphi_{Ieo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Iee}\left( {{n + 1},\omega_{2}} \right)}} + \left\{ {{\varphi_{Iee}\left( {n,\omega_{2}} \right)} + {\varphi_{Iee}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Iee}\left( {{n + 1},\omega_{2}} \right)}} - \left\{ {{\varphi_{Iee}\left( {n,\omega_{2}} \right)} + {\varphi_{Iee}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{- {\varphi_{Ioo}\left( {n,\omega_{2}} \right)}} + {\varphi_{Ioo}\left( {{n + 2},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1352) \\ {{I_{2{AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Ieo}\left( {n,\omega_{2}} \right)} + {\varphi_{Ieo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Ioe}\left( {n,\omega_{2}} \right)} - {\varphi_{Ioe}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Ioo}\left( {{n + 1},\omega_{2}} \right)}} + \left\{ {{\varphi_{Ioo}\left( {n,\omega_{2}} \right)} + {\varphi_{Ioo}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{2{{\mu\varphi}_{Ioo}\left( {{n + 1},\omega_{2}} \right)}} - \left\{ {{\varphi_{Ioo}\left( {n,\omega_{2}} \right)} + {{\varphi I}_{oo}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Iee}\left( {n,\omega_{2}} \right)} - {\varphi_{Iee}\left( {{n + 2},\omega_{2}} \right)}} \right\}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1353) \end{matrix}$

Though a boundary condition has not been reflected yet, they satisfy the primal simultaneous differential equations (1087), (1088) with respect to an arbitrary n.

11.4.24 Solution Function Using Modified Bessel Function of the Second Kind

Eigenvalues ω₁, ω₂ are collectively represented as ω, and solutions using the modified Bessel function of the second kind K_(m) of eqs. (1108), (1112) are expressed as follows according to eq. (1209):

[Formula 556]

φ_(Kee)(r,θ,n,ω)≡cos 2nθ·K _(2n)(ωr)  (1354)

φ_(Koo)(r,θ,n,ω)≡sin 2nθ·K _(2n)(ωr)  (1355)

φ_(Koe)(r,θ,n,ω)≡cos(2n+1)θ·K _(2n+1)(ωr)  (1356)

φ_(Keo)(r,θ,n,ω)≡sin(2n+1)θ·K _(2n+1)(ωr)  (1357)

Master variables of these functions are r and θ, while n and ω represent parameters. As the equation is long if all of these variables are described, only variables that should be noticed are described hereinafter. For example, in the case where n changes on both sides of an equation but r, θ, ω do not change, the argument is expressed as (n). Here, derived functions as follows are obtained:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 557} \right\rbrack & \; \\ {{\frac{\partial}{\partial x}{\varphi_{Kee}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Koe}\left( {n - 1} \right)} + {\varphi_{Koe}(n)}} \right\}}} & (1358) \\ {{\frac{\partial}{\partial x}{\varphi_{Koo}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Keo}\left( {n - 1} \right)} + {\varphi_{Keo}(n)}} \right\}}} & (1359) \\ {{\frac{\partial}{\partial x}{\varphi_{Keo}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Koo}(n)} + {\varphi_{Koo}\left( {n + 1} \right)}} \right\}}} & (1360) \\ {{{\frac{\partial}{\partial x}{\varphi_{Koe}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Kee}(n)} + {\varphi_{Kee}\left( {n + 1} \right)}} \right\}}}{{and},}} & (1361) \\ {{\frac{\partial}{\partial y}{\varphi_{Kee}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Keo}\left( {n - 1} \right)} - {\varphi_{Keo}(n)}} \right\}}} & (1362) \\ {{\frac{\partial}{\partial y}{\varphi_{Koo}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Koe}\left( {n - 1} \right)} - {\varphi_{Koe}(n)}} \right\}}} & (1363) \\ {{\frac{\partial}{\partial y}{\varphi_{Keo}(n)}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Kee}(n)} - {\varphi_{Kee}\left( {n + 1} \right)}} \right\}}} & (1364) \\ {{\frac{\partial}{\partial y}{\varphi_{Koe}(n)}} = {\frac{\omega}{2}\left\{ {{\varphi_{Koo}(n)} - {\varphi_{Koo}\left( {n + 1} \right)}} \right\}}} & (1365) \end{matrix}$

Further, the following relationship equations are derived from eqs. (1358) to (1361):

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 558} \right\rbrack} & \; \\ {\mspace{79mu} {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Kee}(n)} - {\varphi_{Kee}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Koe}\left( {n - 1} \right)} - {\varphi_{Koe}\left( {n + 1} \right)}} \right\}}}} & (1366) \\ {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Koo}(n)} - {\varphi_{Koo}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Keo}\left( {n - 1} \right)} - {\varphi_{Keo}\left( {n + 1} \right)}} \right\}}} & (1367) \\ {\mspace{79mu} {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Keo}(n)} - {\varphi_{Keo}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Koo}(n)} - {\varphi_{Koo}\left( {n + 2} \right)}} \right\}}}} & (1368) \\ {\mspace{79mu} {{\frac{\partial}{\partial x}\left\{ {{\varphi_{Koe}(n)} - {\varphi_{Koe}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Kee}(n)} - {\varphi_{Kee}\left( {n + 2} \right)}} \right\}}}} & (1369) \end{matrix}$

The following relationship equations are derived from eqs. (1362) to (1365):

$\begin{matrix} {\mspace{79mu} {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Kee}(n)} + {\varphi_{Kee}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Keo}\left( {n - 1} \right)} - {\varphi_{Keo}\left( {n + 1} \right)}} \right\}}}} & (1370) \\ {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Koo}(n)} + {\varphi_{Koo}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Koe}\left( {n - 1} \right)} - {\varphi_{Koe}\left( {n + 1} \right)}} \right\}}} & (1371) \\ {\mspace{79mu} {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Keo}(n)} + {\varphi_{Keo}\left( {n + 1} \right)}} \right\}} = {{- \frac{\omega}{2}}\left\{ {{\varphi_{Kee}(n)} - {\varphi_{Kee}\left( {n + 2} \right)}} \right\}}}} & (1372) \\ {\mspace{79mu} {{\frac{\partial}{\partial y}\left\{ {{\varphi_{Koe}(n)} + {\varphi_{Koe}\left( {n + 1} \right)}} \right\}} = {\frac{\omega}{2}\left\{ {{\varphi_{Koo}(n)} - {\varphi_{Koo}\left( {n + 2} \right)}} \right\}}}} & (1373) \end{matrix}$

Here, with focus on the combinations of eqs. (1366) and (1371), eqs. (1367) and (1370), eqs. (1369) and (1372), and eqs. (1368) and (1373), the following four combinations are defined:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 559} \right\rbrack & \; \\ {{\varphi_{K\; 1\; {SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Kee}(n)}} + {\varphi_{Kee}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Koo}(n)} + {\varphi_{Koo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1374) \\ {{\varphi_{K\; 1\; {AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Koo}(n)} - {\varphi_{Koo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Kee}(n)} + {\varphi_{Kee}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1375) \\ {{\varphi_{K\; 1\; {SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Koe}(n)}} + {\varphi_{Koe}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Keo}(n)} + {\varphi_{Keo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1376) \\ {{\varphi_{K\; 1\; {AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Keo}(n)} - {\varphi_{Keo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Koe}(n)} + {\varphi_{Koe}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1377) \end{matrix}$

Then, we find that the following eq. (1120) is established for each of the combinations:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 560} \right\rbrack & \; \\ {{{\frac{\partial}{\partial x}\varphi_{x}} + {\frac{\partial}{\partial y}\varphi_{y}}} = 0} & \underset{({Aforementioned})}{(1120)} \end{matrix}$

Therefore, we can see that the combinations of eqs. (1374) to (1377) in which ω is replaced with ω₁ of eq. (1099) satisfy the primal simultaneous eigenvalue problems of eqs. (1092) and (1093). In other words, ω_(x), φ_(y) are primal eigenfunctions. Further, according to eqs. (1157), (1158), the same functions as φ_(x), φ_(y), are dual eigenfunctions φ_(x)*, φ_(y)*. Thus, four sets of the functions φ_(x), φ_(y), φ_(x)*, φ_(y)* that satisfy the primal simultaneous differential equations (1087) and (1088) are obtained.

With use of eqs. (74) to (76), stresses can be calculated by eqs. (1374) to (1377) as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 561} \right\rbrack & \; \\ {{\varphi_{K\; 1\; {SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {G\; \omega \left\{ {{\varphi_{Koe}\left( {n - 1} \right)} - {\varphi_{Koe}\left( {n + 1} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega \left\{ {{\varphi_{Keo}\left( {n - 1} \right)} + {\varphi_{Keo}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1378) \\ {{\varphi_{K\; 1\; {AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {G\; \omega \left\{ {{- {\varphi_{Keo}\left( {n - 1} \right)}} + {\varphi_{Keo}\left( {n + 1} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega \left\{ {{\varphi_{Koe}\left( {n - 1} \right)} + {\varphi_{Koe}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1379) \\ {{\varphi_{K\; 1\; {SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {G\; \omega \left\{ {{\varphi_{Kee}(n)} - {\varphi_{Kee}\left( {n + 2} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega \left\{ {{\varphi_{Koo}(n)} + {\varphi_{Koo}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1380) \\ {{\varphi_{K\; 1\; {AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {G\; \omega \left\{ {{- {\varphi_{Koo}(n)}} + {\varphi_{Koo}\left( {n + 2} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega \left\{ {{\varphi_{Kee}(n)} + {\varphi_{Kee}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1381) \end{matrix}$

In this way, the combinations in which ω is replaced with ω₁ of eq. (1099) express stresses σ_(x), σ_(y), τ_(xy) based on the primal eigenfunctions. It should be noted that according to eqs. (82) to (84) and eqs. (1157), (1158), the same as σ_(x), σ_(y), τ_(xy) represent stresses σ_(x)*, σ_(y)*, τ_(xy)* based on the dual eigenfunctions. Thus, four sets of the functions expressing stresses σ_(x), σ_(y), τ_(xy), σ_(x)*, σ_(y)*, τ_(xy)* are obtained.

The four sets are defined again below:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 562} \right\rbrack & \; \\ {{K_{1\; {SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Kee}\left( {n,\omega_{1}} \right)}} + {\varphi_{Kee}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Koo}\left( {n,\omega_{1}} \right)} + {\varphi_{Koo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Koe}\left( {{n - 1},\omega_{1}} \right)} - {\varphi_{Koe}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{\varphi_{Keo}\left( {{n - 1},\omega_{1}} \right)} + {\varphi_{Keo}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {{{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}}\;} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1382) \\ {{K_{1\; {AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Koo}\left( {n,\omega_{1}} \right)} - {\varphi_{Koo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Kee}\left( {n,\omega_{1}} \right)} + {\varphi_{Kee}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{x} \equiv {G\; \omega_{1}\left\{ {{- {\varphi_{Keo}\left( {{n - 1},\omega_{1}} \right)}} + {\varphi_{Keo}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{\varphi_{Koe}\left( {{n - 1},\omega_{1}} \right)} + {\varphi_{Koe}\left( {{n + 1},\omega_{1}} \right)}} \right\}}} \\ {{{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}}\;} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1383) \\ {{K_{1\; {SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{- {\varphi_{Koe}\left( {n,\omega_{1}} \right)}} + {\varphi_{Koe}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Keo}\left( {n,\omega_{1}} \right)} + {\varphi_{Keo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{x} \equiv {G\; \omega_{1}\left\{ {{\varphi_{Kee}\left( {n,\omega_{1}} \right)} - {\varphi_{Kee}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{\varphi_{Koo}\left( {n,\omega_{1}} \right)} + {\varphi_{Koo}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {{{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}}\;} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1384) \\ \left\lbrack {{Formula}\mspace{14mu} 563} \right\rbrack & \; \\ {{K_{1\; {AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Keo}\left( {n,\omega_{1}} \right)} - {\varphi_{Keo}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Koe}\left( {n,\omega_{1}} \right)} + {\varphi_{Koe}\left( {{n + 1},\omega_{1}} \right)}}} \\ {\varphi_{x} \equiv {G\; \omega_{1}\left\{ {{- {\varphi_{Koo}\left( {n,\omega_{1}} \right)}} + {\varphi_{Koo}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{\varphi_{Kee}\left( {n,\omega_{1}} \right)} + {\varphi_{Kee}\left( {{n + 2},\omega_{1}} \right)}} \right\}}} \\ {{{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}}\;} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1385) \end{matrix}$

Though a boundary condition has not been reflected yet, they satisfy the primal simultaneous differential equations (1087), (1088) with respect to an arbitrary n.

Similarly, focusing on combinations of eqs. (1370) and (1367), eqs. (1371) and (1366), eqs. (1373) and (1368), and eqs. (1372) and (1369), we define the following four combinations:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 564} \right\rbrack & \; \\ {{\varphi_{K\; 2\; {SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Kee}(n)} + {\varphi_{Kee}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Koo}(n)}} + {\varphi_{Koo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1386) \\ {{\varphi_{K\; 2\; {SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Koo}(n)} + {\varphi_{Koo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Kee}(n)} - {\varphi_{Kee}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1387) \\ {{\varphi_{K\; 2\; {SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Koe}(n)} + {\varphi_{Koe}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Keo}(n)}} + {\varphi_{Keo}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1388) \\ {{\varphi_{K\; 2\; {AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Keo}(n)} + {\varphi_{Keo}\left( {n + 1} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Koe}(n)} - {\varphi_{Koe}\left( {n + 1} \right)}}} \end{matrix} \right.} & (1389) \end{matrix}$

Regarding each of the same, the following eq. (1128) is established:

$\begin{matrix} {{{\frac{\partial}{\partial x}\varphi_{y}} - {\frac{\partial}{\partial y}\varphi_{x}}} = 0} & \underset{({Aforementioned})}{(1128)} \end{matrix}$

Therefore, we can see that the combinations of eqs. (1386) to (1389) in which ω is replaced with ω₂ of eq. (1100) satisfy eqs. (1092) and (1093) of the primal simultaneous eigenvalue problem. In other words, φ_(x), φ_(y), are primal eigenfunctions. Further, according to eqs. (1173), (1174), the same functions as φ_(x), φ_(y) are dual eigenfunctions φ_(x)*, φ_(y)*. Thus, four sets of the functions φ_(x), φ_(y), φ_(x)*, φ_(y)* that satisfy the primal simultaneous differential equations (1087) and (1088) are obtained.

With use of eqs. (74) to (76), stresses can be calculated by eqs. (1386) to (1389) as follows:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 565} \right\rbrack} & \; \\ {{\sigma_{K\; 2\; {SA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{{- 2}{{\mu\varphi}_{Koe}(n)}} - \left\{ {{\varphi_{Koe}\left( {n - 1} \right)} + {\varphi_{Koe}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{{- 2}{{\mu\varphi}_{Koe}(n)}} + \left\{ {{\varphi_{Koe}\left( {n - 1} \right)} + {\varphi_{Koe}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Keo}\left( {n - 1} \right)} - {\varphi_{Keo}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1390) \\ {{\sigma_{K\; 2\; {AS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{{- 2}{{\mu\varphi}_{Keo}(n)}} - \left\{ {{\varphi_{Keo}\left( {n - 1} \right)} + {\varphi_{Keo}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{{- 2}{{\mu\varphi}_{Keo}(n)}} + \left\{ {{\varphi_{Keo}\left( {n - 1} \right)} + {\varphi_{Keo}\left( {n + 1} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{- {\varphi_{Koe}\left( {n - 1} \right)}} + {\varphi_{Koe}\left( {n + 1} \right)}} \right\}}} \end{matrix} \right.} & (1391) \\ {{\sigma_{K\; 2\; {SS}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{{- 2}{{\mu\varphi}_{Kee}\left( {n + 1} \right)}} - \left\{ {{\varphi_{Kee}(n)} + {\varphi_{Kee}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{{- 2}{{\mu\varphi}_{Kee}\left( {n + 1} \right)}} + \left\{ {{\varphi_{Kee}(n)} + {\varphi_{Kee}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Koo}(n)} - {\varphi_{Koo}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1392) \\ {{\sigma_{K\; 2\; {AA}}(n)} \equiv \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; {\omega \left\lbrack {{{- 2}{{\mu\varphi}_{Koo}\left( {n + 1} \right)}} - \left\{ {{\varphi_{Koo}(n)} + {\varphi_{Koo}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega \left\lbrack {{{- 2}{{\mu\varphi}_{Koo}\left( {n + 1} \right)}} + \left\{ {{\varphi_{Koo}(n)} + {\varphi_{Koo}\left( {n + 2} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega \left\{ {{\varphi_{Kee}(n)} + {\varphi_{Kee}\left( {n + 2} \right)}} \right\}}} \end{matrix} \right.} & (1393) \end{matrix}$

In this way, the combinations in which ω is replaced with ω₂ of eq. (1100) express stresses σ_(x), σ_(y), τ_(xy) based on the primal eigenfunctions. It should be noted that according to eqs. (82) to (84) and eqs. (1173), (1174), the same as σ_(x), σ_(y), τ_(xy) represent stresses σ_(x)*, σ_(y)*, τ_(xy)* based on the dual eigenfunctions. Thus, four sets of the functions expressing stresses σ_(x), σ_(y), τ_(xy), σ_(x)*, σ_(y)*, τ_(xy)* are obtained.

The four sets are defined again below:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 566} \right\rbrack} & \; \\ {{K_{2\; {SA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Kee}\left( {n,\omega_{2}} \right)} + {\varphi_{Kee}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Koo}\left( {n,\omega_{2}} \right)} + {\varphi_{Koo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\sigma_{x} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}{{\mu\varphi}_{Koe}\left( {n,\omega_{2}} \right)}} - \left\{ {{\varphi_{Koe}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Koe}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}{{\mu\varphi}_{Koe}\left( {n,\omega_{2}} \right)}} + \left\{ {{\varphi_{Koe}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Koe}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Keo}\left( {{n - 1},\omega_{2}} \right)} - {\varphi_{Keo}\left( {{n + 1},\omega_{2}} \right)}} \right\}}} \\ {{{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}}\;} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1394) \\ {{K_{2\; {AS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Koo}\left( {n,\omega_{2}} \right)} + {\varphi_{Koo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Kee}\left( {n,\omega_{2}} \right)} - {\varphi_{Kee}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{x} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}{{\mu\varphi}_{Keo}\left( {n,\omega_{2}} \right)}} - \left\{ {{\varphi_{Keo}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Keo}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}{{\mu\varphi}_{Keo}\left( {n,\omega_{2}} \right)}} + \left\{ {{\varphi_{Keo}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Keo}\left( {{n + 1},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Koe}\left( {{n - 1},\omega_{2}} \right)} + {\varphi_{Koe}\left( {{n + 1},\omega_{2}} \right)}} \right\}}} \\ {{{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}}\;} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1395) \\ {{K_{2\; {SS}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Koe}\left( {n,\omega_{2}} \right)} + {\varphi_{Koe}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{- {\varphi_{Keo}\left( {n,\omega_{2}} \right)}} + {\varphi_{Keo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{x} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}{{\mu\varphi}_{Kee}\left( {{n + 1},\omega_{2}} \right)}} - \left\{ {{\varphi_{Kee}\left( {n,\omega_{2}} \right)} + {\varphi_{Kee}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}{{\mu\varphi}_{Kee}\left( {{n + 1},\omega_{2}} \right)}} + \left\{ {{\varphi_{Kee}\left( {n,\omega_{2}} \right)} + {\varphi_{Kee}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{\varphi_{Koo}\left( {n,\omega_{2}} \right)} - {\varphi_{Koo}\left( {{n + 2},\omega_{2}} \right)}} \right\}}} \\ {{{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}}\;} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1396) \\ {{K_{2\; {AA}}(n)} \equiv \left\{ \begin{matrix} {\varphi_{x} \equiv {{\varphi_{Keo}\left( {n,\omega_{2}} \right)} + {\varphi_{Keo}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{y} \equiv {{\varphi_{Koe}\left( {n,\omega_{2}} \right)} - {\varphi_{Koe}\left( {{n + 1},\omega_{2}} \right)}}} \\ {\varphi_{x} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}{{\mu\varphi}_{Koo}\left( {{n + 1},\omega_{2}} \right)}} - \left\{ {{\varphi_{Koo}\left( {n,\omega_{2}} \right)} + {\varphi_{Koo}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\sigma_{y} \equiv {G\; {\omega_{2}\left\lbrack {{{- 2}{{\mu\varphi}_{Koo}\left( {{n + 1},\omega_{2}} \right)}} + \left\{ {{\varphi_{Koo}\left( {n,\omega_{2}} \right)} + {\varphi_{Koo}\left( {{n + 2},\omega_{2}} \right)}} \right\}} \right\rbrack}}} \\ {\tau_{xy} \equiv {G\; \omega_{2}\left\{ {{- {\varphi_{Kee}\left( {n,\omega_{2}} \right)}} + {\varphi_{Kee}\left( {{n - 2},\omega_{2}} \right)}} \right\}}} \\ {{{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}}\;} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right.} & (1397) \end{matrix}$

Though a boundary condition has not been reflected yet, they satisfy the primal simultaneous differential equations (1087), (1088) with respect to an arbitrary n.

11.4.25 Function Set of Primal Eigenfunction and Dual Eigenfunction

By imposing a boundary condition on the combination of function sets shown in Sections 11.4.21 to 11.4.24, eigenvalues and eigenfunctions belonging to the same are settled. Regarding each of four deformation types of the modes SA, AS, SS, and AA, specific formulae of the primal eigenfunctions φ_(x), φ_(y), the stresses σ_(x), σ_(y), τ_(xy) based on these primal eigenfunctions, the dual eigenfunctions φ_(x)*, φ_(y)*, and the stresses σ_(x)*, σ_(y)*, τ_(xy)* based on these dual eigenfunctions are show below.

[SA]

To the mode SA, the following eight sets in total belong: J_(1SA) of eq. (1250); J_(2SA) of eq. (1262); Y_(1SA) of eq. (1294); Y_(2SA) of eq. (1306); I_(1SA) of eq. (1338); I_(2SA) of eq. (1350); K_(1SA) of eq. (1382); and K_(2SA) of eq. (1394). As they serve as solution functions with respect to an arbitrary integer n, the total number is 8n.

(1) Regarding J_(1SA), we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 567} \right\rbrack} & \; \\ {\mspace{79mu} \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{J_{2\; n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{J_{{2\; n} + 2}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{- {J_{2\; n}\left( {\omega_{1}r} \right)}} \cdot \sin}\; 2\; n\; \theta} + {{{J_{{2\; n} + 2}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},\mspace{14mu} {\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right.} & (1398) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{{{J_{{2n} - 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} - 1} \right)}}\theta} - {{{J_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{{{J_{{2n} - 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} - 1} \right)}}\theta} + {{{J_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},\mspace{14mu} {\sigma_{y}^{*} \equiv {- \sigma_{y}}},\mspace{14mu} {\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1399) \end{matrix}$

Converting eq. (1398) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 568} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{J_{2\; n}\left( {\omega_{1}r} \right)} + {J_{{2\; n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{- {J_{2\; n}\left( {\omega_{1}r} \right)}} + {J_{{2\; n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1400) \end{matrix}$

Converting eq. (1399) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{J_{{2n} - 1}\left( {\omega_{1}r} \right)} - {J_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {{- G}\; \omega_{1}\left\{ {{J_{{2n} - 1}\left( {\omega_{1}r} \right)} + {J_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1401) \end{matrix}$

(2) Regarding J_(2SA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 569} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{- {J_{2\; n}\left( {\omega_{2}r} \right)}} \cdot \cos}\; 2\; n\; \theta} + {{{J_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{J_{2\; n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{J_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},\mspace{14mu} {\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1402) \\ \left\{ \begin{matrix} {g_{1} \equiv {2{\mu \cdot {J_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta}} \\ {g_{2} \equiv {{{{J_{{2\; n} - 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} - 1} \right)}}\theta} + {{{J_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {g_{3} \equiv {{{{J_{{2\; n} - 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} - 1} \right)}}\theta} - {{{J_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},\mspace{14mu} {\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},\mspace{14mu} {\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},\mspace{14mu} {\sigma_{y}^{*} \equiv {- \sigma_{y}}},\mspace{20mu} {\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1403) \end{matrix}$

Converting eq. (1402) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 570} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{- {J_{2\; n}\left( {\omega_{2}r} \right)}} + {J_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{J_{2\; n}\left( {\omega_{2}r} \right)} + {J_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1404) \end{matrix}$

Converting eq. (1403) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {2\; {\mu \cdot {J_{{2n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 1} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{J_{{2n} - 1}\left( {\omega_{2}r} \right)} + {J_{{2n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2n} + 1} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{J_{{2n} - 1}\left( {\omega_{2}r} \right)} - {J_{{2n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2n} + 1} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1}h_{2}} \right)}}},{\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},{\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},{\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},{\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1405) \end{matrix}$

(3) Regarding Y_(1SA), we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 571} \right\rbrack} & \; \\ {\mspace{79mu} \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{Y_{2n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2n\; \theta} + {{{Y_{{2n} + 2}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{- {Y_{2n}\left( {\omega_{1}r} \right)}} \cdot \sin}\; 2n\; \theta} + {{{Y_{{2n} + 2}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 2} \right)}}\theta}}} \\ {{\varphi_{x} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right.} & (1406) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{{{Y_{{2n} - 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} - 1} \right)}}\theta} - {{{Y_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 3} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{{{Y_{{2n} - 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} - 1} \right)}}\theta} + {{{Y_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 3} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1407) \end{matrix}$

Converting eq. (1406) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 571} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {\left\{ {{Y_{2n}\left( {\omega_{1}r} \right)} + {Y_{{2n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2n} + 1} \right)}\theta}} \\ {\varphi_{y} \equiv {\left\{ {{- {Y_{2n}\left( {\omega_{1}r} \right)}} + {Y_{{2n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},{\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1408) \end{matrix}$

Converting eq. (1407) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{Y_{{2n} - 1}\left( {\omega_{1}r} \right)} - {Y_{{2n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2n} + 1} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {{- G}\; \omega_{1}\left\{ {{Y_{{2n} - 1}\left( {\omega_{1}r} \right)} + {Y_{{2n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2n} + 1} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},{\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},{\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1409) \end{matrix}$

(4) Regarding Y_(2SA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 573} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{- {Y_{2n}\left( {\omega_{2}r} \right)}} \cdot \cos}\; 2n\; \theta} + {{{Y_{{2n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{Y_{2n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{Y_{{2n}\; + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1410) \\ \left\lbrack {{Formula}\mspace{14mu} 574} \right\rbrack & \; \\ \left\{ \begin{matrix} {g_{1} \equiv {2\; {\mu \cdot {Y_{{2n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 1} \right)}}\theta}} \\ {g_{2} \equiv {{{{Y_{{2n} - 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} - 1} \right)}}\theta} + {{{Y_{{2n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 3} \right)}}\theta}}} \\ {g_{3} \equiv {{{{Y_{{2n} - 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} - 1} \right)}}\theta} - {{{Y_{{2n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 3} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},{\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},{\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*\;} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}^{*}}}} \end{matrix} \right. & (1411) \end{matrix}$

Converting eq. (1410) according to eqs. (969), (970) gives:

$\begin{matrix} \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{- {Y_{2n}\left( {\omega_{2}r} \right)}} + {Y_{{2n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{Y_{2n}\left( {\omega_{2}r} \right)} + {Y_{{2n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},{\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1412) \end{matrix}$

Converting eq. (1411) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 575} \right\rbrack & \; \\ \left\{ \begin{matrix} {h_{1} \equiv {2\; {\mu \cdot {Y_{{2n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 1} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{Y_{{2n} - 1}\left( {\omega_{2}r} \right)} + {Y_{{2n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2n} + 1} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{Y_{{2n} - 1}\left( {\omega_{2}r} \right)} - {Y_{{2n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2n} + 1} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},{\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},{\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},{\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},{\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1413) \end{matrix}$

(5) Regarding I_(1SA), we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 576} \right\rbrack} & \; \\ {\mspace{79mu} \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{- {I_{2n}\left( {\omega_{1}r} \right)}} \cdot \cos}\; 2n\; \theta} + {{{I_{{2n} + 2}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{I_{2n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2n\; \theta} + {{{I_{{2n} + 2}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right.} & (1414) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{{{- {I_{{2n} - 1}\left( {\omega_{1}r} \right)}} \cdot {\cos \left( {{2n} - 1} \right)}}\theta} + {{{I_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 3} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{{{I_{{2n} - 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} - 1} \right)}}\theta} + {{{I_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 3} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1415) \end{matrix}$

Converting eq. (1414) according to eqs. (969), (970) gives:

$\begin{matrix} \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{- {I_{2n}\left( {\omega_{1}r} \right)}} + {I_{{2n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{I_{2n}\left( {\omega_{1}r} \right)} + {I_{{2n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},{\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1416) \end{matrix}$

Converting eq. (1415) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 577} \right\rbrack & \; \\ \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{- {I_{{2n} - 1}\left( {\omega_{1}r} \right)}} + {I_{{2n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2n} + 1} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {G\; \omega_{1}\left\{ {{I_{{2n} - 1}\left( {\omega_{1}r} \right)} + {I_{{2n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2n} + 1} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},{\sigma_{\theta}^{*} \equiv \sigma_{\theta}},{\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1417) \end{matrix}$

(6) Regarding I_(2SA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 578} \right. & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{I_{2n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2n\; \theta} + {{{I_{{2n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{- {I_{2n}\left( {\omega_{2}r} \right)}} \cdot \sin}\; 2\; n\; \theta} + {{{I_{{2n}\; + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1418) \\ \left\{ \begin{matrix} {g_{1} \equiv {2\; {\mu \cdot {I_{{2n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 1} \right)}}\theta}} \\ {g_{2} \equiv {{{{I_{{2n} - 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} - 1} \right)}}\theta} + {{{I_{{2n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 3} \right)}}\theta}}} \\ {g_{3} \equiv {{{{I_{{2n} - 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} - 1} \right)}}\theta} + {{{I_{{2n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 3} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},{\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},{\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*\;} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}^{*}}} \end{matrix} \right. & (1419) \end{matrix}$

Converting eq. (1418) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 579} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{I_{2n}\left( {\omega_{2}r} \right)} + {I_{{2n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{- {I_{2n}\left( {\omega_{2}r} \right)}} + {I_{{2n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2n} +} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},{\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1420) \end{matrix}$

Converting eq. (1419) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {2\; {\mu \cdot {I_{{2n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 1} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{I_{{2n} - 1}\left( {\omega_{2}r} \right)} + {I_{{2n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2n} + 1} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{- {I_{{2n} - 1}\left( {\omega_{2}r} \right)}} + {I_{{2n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2n} + 1} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},{\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},{\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},{\sigma_{\theta}^{*} \equiv \sigma_{\theta}},{\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1421) \end{matrix}$

(7) Regarding K_(1SA), we obtain:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 580} \right\rbrack} & \; \\ {\mspace{79mu} \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{- {K_{2n}\left( {\omega_{1}r} \right)}} \cdot \cos}\; 2n\; \theta} + {{{K_{{2n} + 2}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{K_{2n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2n\; \theta} + {{{K_{{2n} + 2}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right.} & (1422) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{{{K_{{2n} - 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} - 1} \right)}}\theta} + {{{K_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 3} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{{{K_{{2n} - 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} - 1} \right)}}\theta} + {{{K_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 3} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1423) \end{matrix}$

Converting eq. (1422) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 581} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{- {K_{2n}\left( {\omega_{1}r} \right)}} + {K_{{2n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{K_{2n}\left( {\omega_{1}r} \right)} + {K_{{2n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},{\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1424) \end{matrix}$

Converting eq. (1423) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{K_{{2\; n} - 1}\left( {\omega_{1}r} \right)} - {K_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {{- G}\; \omega_{1}\left\{ {{K_{{2\; n} - 1}\left( {\omega_{1}r} \right)} + {K_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \sigma_{\theta}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1425) \end{matrix}$

(8) Regarding K_(2SA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 582} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{K_{2\; n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{K_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{- {K_{2\; n}\left( {\omega_{2}r} \right)}} \cdot \sin}\; 2\; n\; \theta} + {{{K_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}},} \end{matrix} \right. & (1426) \\ \left\{ \begin{matrix} {g_{1} \equiv {{- 2}\; {\mu \cdot {K_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta}} \\ {g_{2} \equiv {{{{K_{{2\; n} - 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} - 1} \right)}}\theta} + {{{K_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {g_{3} \equiv {{{{K_{{2\; n} - 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} - 1} \right)}}\theta} - {{{K_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},\mspace{14mu} {\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},\mspace{14mu} {\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv \sigma_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1427) \end{matrix}$

[Formula 583]

Converting eq. (1426) according to eqs. (969), (970) gives:

$\begin{matrix} \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{K_{2\; n}\left( {\omega_{2}r} \right)} + {K_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{- {K_{2\; n}\left( {\omega_{2}r} \right)}} + {K_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1428) \end{matrix}$

Converting eq. (1427) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {{- 2}\; {\mu \cdot {K_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{K_{{2\; n} - 1}\left( {\omega_{2}r} \right)} + {K_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{K_{{2\; n} - 1}\left( {\omega_{2}r} \right)} - {K_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},\mspace{14mu} {\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},\mspace{14mu} {\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \sigma_{\theta}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1429) \end{matrix}$

[AS]

To the mode AS, the following eight sets in total belong: J_(1AS) of eq. (1251); J_(2AS) of eq. (1263); Y_(1AS) of eq. (1295); Y_(2AS) of eq. (1307); I_(1AS) of eq. (1339); I_(2AS) of eq. (1351); K_(1AS) of eq. (1383); and K_(2AS) of eq. (1395). As they serve as solution functions with respect to an arbitrary integer n, the total number is 8n.

(1) Regarding J_(1AS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 584} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{J_{2\; n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{J_{{2\; n} + 2}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{J_{2\; n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2\; n\; \theta} - {{{J_{{2\; n} + 2}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},\mspace{14mu} {\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1430) \\ \left\lbrack {{Formula}\mspace{14mu} 585} \right\rbrack & \; \\ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\begin{Bmatrix} {{{{J_{{2\; n} - 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} - 1} \right)}}\theta} -} \\ {{{J_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta} \end{Bmatrix}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\begin{Bmatrix} {{{{J_{{2\; n} - 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} - 1} \right)}}\theta} +} \\ {{{J_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta} \end{Bmatrix}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},\mspace{14mu} {\sigma_{y}^{*} \equiv {- \sigma_{y}}},\mspace{14mu} {\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} & (1431) \end{matrix}$

Converting eq. (1430) according to eqs. (969), (970) gives:

$\begin{matrix} \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{J_{2\; n}\left( {\omega_{1}r} \right)} + {J_{{2\; n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{J_{2\; n}\left( {\omega_{\; 1}r} \right)} - {J_{{2\; n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}},} \end{matrix} \right. & (1432) \end{matrix}$

Converting eq. (1431) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 586} \right\rbrack & \; \\ \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{J_{{2\; n} - 1}\left( {\omega_{1}r} \right)} - {J_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {G\; \omega_{1}\left\{ {{J_{{2\; n} - 1}\left( {\omega_{1}r} \right)} + {J_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1433) \end{matrix}$

(2) Regarding J_(2AS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 587} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{J_{2\; n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2\; n\; \theta} - {{{J_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{J_{2\; n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{J_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},\mspace{14mu} {\varphi_{y}^{*} \equiv {- \varphi_{y}}},} \end{matrix} \right. & (1434) \\ \left\{ \begin{matrix} {g_{1} \equiv {{- 2}\; {\mu \cdot {J_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta}} \\ {g_{2} \equiv {{{{J_{{2\; n} - 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} - 1} \right)}}\theta} + {{{J_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {g_{3} \equiv {{{{J_{{2\; n} - 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} - 1} \right)}}\theta} - {{{J_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},\mspace{14mu} {\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},\mspace{14mu} {\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},\mspace{14mu} {\sigma_{y}^{*} \equiv {- \sigma_{y}}},\mspace{14mu} {\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1435) \end{matrix}$

Converting eq. (1434) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 588} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{J_{2\; n}\left( {\omega_{2}r} \right)} - {J_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{J_{2\; n}\left( {\omega_{\; 2}r} \right)} + {J_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1436) \end{matrix}$

Converting (1435) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {{- 2}\; {\mu \cdot {J_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{J_{{2\; n} - 1}\left( {\omega_{2}r} \right)} + {J_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{J_{{2\; n} - 1}\left( {\omega_{2}r} \right)} - {J_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},\mspace{14mu} {\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},\mspace{14mu} {\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1437) \end{matrix}$

(3) Regarding Y_(1AS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 589} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{Y_{2\; n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{Y_{{2\; n} + 2}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{Y_{2\; n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2\; n\; \theta} - {{{Y_{{2\; n} + 2}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},\mspace{14mu} {\varphi_{y}^{*} \equiv {- \varphi_{y}}},} \end{matrix} \right. & (1438) \\ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\begin{Bmatrix} {{{{Y_{{2\; n} - 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} - 1} \right)}}\theta} -} \\ {{{Y_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta} \end{Bmatrix}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\begin{Bmatrix} {{{{Y_{{2\; n} - 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} - 1} \right)}}\theta} +} \\ {{{Y_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta} \end{Bmatrix}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},\mspace{14mu} {\sigma_{y}^{*} \equiv {- \sigma_{y}}},\mspace{14mu} {\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} & (1439) \end{matrix}$

Converting eq. (1438) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 590} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{Y_{2\; n}\left( {\omega_{1}r} \right)} + {Y_{{2\; n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{Y_{2\; n}\left( {\omega_{\; 1}r} \right)} - {Y_{{2\; n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}},} \end{matrix} \right. & (1440) \end{matrix}$

Converting (1439) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{Y_{{2\; n} - 1}\left( {\omega_{1}r} \right)} - {Y_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {G\; \omega_{1}\left\{ {{Y_{{2\; n} - 1}\left( {\omega_{1}r} \right)} + {Y_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1441) \end{matrix}$

(4) Regarding Y_(2AS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 591} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{Y_{2\; n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2\; n\; \theta} - {{{Y_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{Y_{2\; n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{Y_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},\mspace{14mu} {\varphi_{y}^{*} \equiv {- \varphi_{y}}},} \end{matrix} \right. & (1442) \\ \left\{ \begin{matrix} {g_{1} \equiv {{- 2}\; {\mu \cdot {Y_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta}} \\ {g_{2} \equiv {{{{Y_{{2\; n} - 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} - 1} \right)}}\theta} + {{{Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {g_{3} \equiv {{{{Y_{{2\; n} - 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} - 1} \right)}}\theta} - {{{Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},\mspace{14mu} {\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},\mspace{14mu} {\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},\mspace{14mu} {\sigma_{y}^{*} \equiv {- \sigma_{y}}},\mspace{14mu} {\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1443) \end{matrix}$

Converting eq. (1442) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 592} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{Y_{2\; n}\left( {\omega_{2}r} \right)} - {Y_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{Y_{2\; n}\left( {\omega_{\; 2}r} \right)} + {Y_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}},} \end{matrix} \right. & (1444) \end{matrix}$

Converting (1443) according to eqs. (977), (978), (979) gives:

$\begin{matrix} {\quad\left\{ \begin{matrix} {h_{1} \equiv {{- 2}\; {\mu \cdot {Y_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta}} \\ {h_{2} \equiv {\begin{Bmatrix} {{Y_{{2\; n} - 1}\left( {\omega_{2}r} \right)} +} \\ {Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \end{Bmatrix}{\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {h_{3} \equiv {\begin{Bmatrix} {{Y_{{2\; n} - 1}\left( {\omega_{2}r} \right)} -} \\ {Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \end{Bmatrix}{\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},\mspace{14mu} {\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},\mspace{14mu} {\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{{\sigma_{r}^{*} \equiv {- \sigma_{r}}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}}\mspace{14mu}} \end{matrix} \right.} & (1445) \end{matrix}$

(5) Regarding I_(1AS), we obtain:

$\begin{matrix} {\; \left\lbrack {{Formula}\mspace{14mu} 593} \right\rbrack} & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{I_{2\; n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2\; n\; \theta} - {{{I_{{2\; n} + 2}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{I_{2\; n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{I_{{2\; n} + 2}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1446) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\begin{Bmatrix} {{{{I_{{2\; n} - 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} - 1} \right)}}\; \theta} -} \\ {{{I_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta} \end{Bmatrix}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\begin{Bmatrix} {{{{I_{{2\; n} - 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} - 1} \right)}}\; \theta} +} \\ {{{I_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta} \end{Bmatrix}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv \sigma_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1447) \end{matrix}$

Converting eq. (1446) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 594} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{I_{2\; n}\left( {\omega_{1}r} \right)} - {I_{{2\; n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {\varphi_{\; \theta} \equiv {\left\{ {{I_{2\; n}\left( {\omega_{1}r} \right)} + {I_{{2\; n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1448) \end{matrix}$

Converting eq. (1447) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{I_{{2\; n} - 1}\left( {\omega_{1}r} \right)} - {I_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {G\; \omega_{1}\left\{ {{I_{{2\; n} - 1}\left( {\omega_{1}r} \right)} + {I_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \sigma_{\theta}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1449) \end{matrix}$

(6) Regarding I_(2AS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 595} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{I_{2\; n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{I_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{I_{2\; n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2\; n\; \theta} - {{{I_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1450) \\ \left\{ \begin{matrix} {g_{1} \equiv {2\; {\mu \cdot {I_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta}} \\ {g_{2} \equiv {{{{I_{{2\; n} - 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} - 1} \right)}}\theta} + {{{I_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {g_{3} \equiv {{{{I_{{2\; n} - 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} - 1} \right)}}\theta} - {{{I_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},\mspace{14mu} {\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},\mspace{14mu} {\tau_{xy}\; \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv \sigma_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1451) \end{matrix}$

Converting eq. (1450) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 596} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{I_{2\; n}\left( {\omega_{2}r} \right)} + {I_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{I_{2\; n}\left( {\omega_{2}r} \right)} - {I_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1452) \end{matrix}$

Converting eq. (1451) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 597} \right\rbrack & \; \\ \left\{ \begin{matrix} {h_{3} \equiv {2\; {\mu \cdot {I_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta}} \\ {h_{1} \equiv {\left\{ {{I_{{2\; n} - 1}\left( {\omega_{2}r} \right)} + {I_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {h_{2} \equiv {\left\{ {{I_{{2\; n} - 1}\left( {\omega_{2}r} \right)} - {I_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},\mspace{14mu} {\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},\mspace{14mu} {\tau_{r\; \theta}\; \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \sigma_{\theta}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1453) \end{matrix}$

(7) Regarding K_(1AS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 598} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{K_{2\; n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2\; n\; \theta} - {{{K_{{2\; n} + 2}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{K_{2\; n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{K_{{2\; n} + 2}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1454) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\begin{Bmatrix} {{{{- {K_{{2\; n} - 1}\left( {\omega_{1}r} \right)}} \cdot {\sin \left( {{2\; n} - 1} \right)}}\theta} +} \\ {{{K_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta} \end{Bmatrix}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\begin{Bmatrix} {{{{K_{{2\; n} - 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} - 1} \right)}}\theta} +} \\ {{{K_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta} \end{Bmatrix}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv \sigma_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1455) \end{matrix}$

Converting eq. (1454) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 599} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{K_{2\; n}\left( {\omega_{1}r} \right)} - {K_{{2\; n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{K_{2\; n}\left( {\omega_{1}r} \right)} + {K_{{2\; n} + 2}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1456) \end{matrix}$

Converting eq. (1455) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{K_{{2\; n} - 1}\left( {\omega_{1}r} \right)} - {K_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {{- G}\; \omega_{1}\left\{ {{K_{{2\; n} - 1}\left( {\omega_{1}r} \right)} + {K_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \sigma_{\theta}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1457) \end{matrix}$

(8) Regarding K_(2AS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 600} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{K_{2\; n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{K_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{K_{2\; n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2\; n\; \theta} - {{{K_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1458) \\ \left\{ \begin{matrix} {g_{1} \equiv {{- 2}\; {\mu \cdot {K_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta}} \\ {g_{2} \equiv {{{{K_{{2\; n} - 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} - 1} \right)}}\theta} + {{{K_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {g_{3} \equiv {{{{- {K_{{2\; n} - 1}\left( {\omega_{2}r} \right)}} \cdot {\cos \left( {{2\; n} - 1} \right)}}\theta} + {{{K_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},\mspace{14mu} {\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},\mspace{14mu} {\tau_{xy}\; \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv \sigma_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1459) \end{matrix}$

Converting eq. (1458) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 601} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{K_{2\; n}\left( {\omega_{2}r} \right)} + {K_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{K_{2\; n}\left( {\omega_{2}r} \right)} - {K_{{2\; n} + 2}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1460) \end{matrix}$

Converting (1459) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {{- 2}\; {\mu \cdot {K_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{K_{{2\; n} - 1}\left( {\omega_{2}r} \right)} + {K_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 1} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{- {K_{{2\; n} - 1}\left( {\omega_{2}r} \right)}} + {K_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 1} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},\mspace{14mu} {\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},\mspace{14mu} {\tau_{r\; \theta}\; \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \sigma_{\theta}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1461) \end{matrix}$

[SS]

To the mode SS, the following eight sets in total belong: J_(1SS) of eq. (1252); J_(2SS) of eq. (1264); Y_(1SS) of eq. (1296); Y_(2SS) of eq. (1308); less of eq. (1340); I_(2SS) of eq. (1352); Kiss of eq. (1384); K_(2SS) of eq. (1396). As they serve as solution functions with respect to an arbitrary integer n, the total number is 8n.

(1) Regarding J_(1SS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 602} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{J_{{2\; n} + 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 1} \right)}}\; \theta} + {{{J_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{- {J_{{2\; n} + 1}\left( {\omega_{1}r} \right)}} \cdot {\sin \left( {{2\; n} + 1} \right)}}\; \theta} + {{{J_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},\mspace{14mu} {\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1462) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{{{J_{2\; n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2\; n\; \theta} - {{{J_{{2\; n} + 4}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 4} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{{{J_{2\; n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{J_{{2\; n} + 4}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 4} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv {- \sigma_{y}}},\mspace{14mu} {\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1463) \end{matrix}$

Converting eq. (1462) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 603} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{J_{{2\; n} + 1}\left( {\omega_{1}r} \right)} + {J_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{- {J_{{2\; n} + 1}\left( {\omega_{1}r} \right)}} + {J_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1464) \end{matrix}$

Converting eq. (1463) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{J_{2\; n}\left( {\omega_{1}r} \right)} - {J_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {{- G}\; \omega_{1}\left\{ {{J_{2\; n}\left( {\omega_{1}r} \right)} + {J_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1465) \end{matrix}$

(2) Regarding J_(2SS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 604} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{- {J_{{2\; n} + 1}\left( {\omega_{2}r} \right)}} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta} + {{{J_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{J_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta} + {{{J_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},\mspace{14mu} {\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1466) \\ \left\{ \begin{matrix} {g_{1} \equiv {2\; {\mu \cdot \; {J_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}} \\ {g_{2} \equiv {{{{J_{2\; n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{J_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 4} \right)}}\theta}}} \\ {g_{3} \equiv {{{{J_{2\; n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2\; n\; \theta} - {{{J_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 4} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},\mspace{14mu} {\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},\mspace{11mu} {\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},\mspace{14mu} {\sigma_{y}^{*} \equiv {- \sigma_{y}}},\mspace{14mu} {\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1467) \end{matrix}$

Converting eq. (1466) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 605} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{- {J_{{2\; n} + 1}\left( {\omega_{2}r} \right)}} + {J_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{J_{{2\; n} + 1}\left( {\omega_{2}r} \right)} + {J_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1468) \end{matrix}$

Converting (1467) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {2\; {\mu \cdot {J_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{J_{2\; n}\left( {\omega_{2}r} \right)} + {J_{{2\; n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{J_{2\; n}\left( {\omega_{2}r} \right)} - {J_{{2\; n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{2} - h_{2}} \right)}}},\mspace{14mu} {\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},\mspace{14mu} {\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1469) \end{matrix}$

(3) Regarding Y_(1SS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 606} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{Y_{{2\; n} + 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta} + {{{Y_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{- {Y_{{2\; n} + 1}\left( {\omega_{1}r} \right)}} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta} + {{{Y_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},\mspace{14mu} {\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1470) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{{{Y_{2\; n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2\; n\; \theta} - {{{Y_{{2\; n} + 4}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 4} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{{{Y_{2\; n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{Y_{{2\; n} + 4}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 4} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},\mspace{14mu} {\sigma_{y}^{*} \equiv {- \sigma_{y}}},\mspace{14mu} {\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1471) \end{matrix}$

Converting eq. (1470) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 607} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{Y_{{2\; n} + 1}\left( {\omega_{1}r} \right)} + {Y_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{- {Y_{{2\; n} + 1}\left( {\omega_{1}r} \right)}} + {Y_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1472) \end{matrix}$

Converting eq. (1471) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 608} \right\rbrack & \; \\ \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{Y_{2\; n}\left( {\omega_{1}r} \right)} - {Y_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {{- G}\; \omega_{1}\left\{ {{Y_{2\; n}\left( {\omega_{1}r} \right)} + {Y_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1473) \end{matrix}$

(4) Regarding Y_(2SS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 609} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{- {Y_{{2\; n} + 1}\left( {\omega_{2}r} \right)}} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta} + {{{Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{Y_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta} + {{{Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},\mspace{14mu} {\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1474) \\ \left\{ \begin{matrix} {g_{1} \equiv {2\; {\mu \; \cdot {Y_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}} \\ {g_{2} \equiv {{{{Y_{2\; n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{Y_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 4} \right)}}\theta}}} \\ {g_{3} \equiv {{{{Y_{2\; n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2\; n\; \theta} - {{{Y_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 4} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},\mspace{14mu} {\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},\mspace{11mu} {\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},\mspace{14mu} {\sigma_{y}^{*} \equiv {- \sigma_{y}}},\mspace{14mu} {\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1475) \end{matrix}$

Converting eqs. (1474) according to eas. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 610} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{- {Y_{{2\; n} + 1}\left( {\omega_{2}r} \right)}} + {Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{Y_{{2\; n} + 1}\left( {\omega_{2}r} \right)} + {Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1476) \end{matrix}$

Converting eq. (1475) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {2\; {\mu \cdot {Y_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{Y_{2\; n}\left( {\omega_{2}r} \right)} + {Y_{{2\; n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{Y_{2\; n}\left( {\omega_{2}r} \right)} - {Y_{{2\; n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{2} - h_{2}} \right)}}},\mspace{14mu} {\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},\mspace{14mu} {\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1477) \end{matrix}$

(5) Regarding I_(1SS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 611} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{- {I_{{2\; n} + 1}\left( {\omega_{1}r} \right)}} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta} + {{{I_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{I_{{2\; n} + 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta} + {{{I_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1478) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{{{- {I_{2\; n}\left( {\omega_{1}r} \right)}} \cdot \cos}\; 2\; n\; \theta} + {{{I_{{2\; n} + 4}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 4} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{{{I_{2\; n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{I_{{2\; n} + 4}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 4} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv \sigma_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1479) \end{matrix}$

Converting eq. (1478) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 612} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{- {I_{{2\; n} + 1}\left( {\omega_{1}r} \right)}} + {I_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{I_{{2\; n} + 1}\left( {\omega_{1}r} \right)} + {I_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1480) \end{matrix}$

Converting eq. (1479) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{- {I_{2\; n}\left( {\omega_{1}r} \right)}} + {I_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {G\; \omega_{1}\left\{ {{I_{2\; n}\left( {\omega_{1}r} \right)} + {I_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \sigma_{\theta}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1481) \end{matrix}$

(6) Regarding I_(2SS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 613} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{I_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta} + {{{I_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{- {I_{{2\; n} + 1}\left( {\omega_{2}r} \right)}} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta} + {{{I_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1482) \\ \left\{ \begin{matrix} {g_{1} \equiv {2\; {\mu \; \cdot {I_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 2} \right)}}\theta}} \\ {g_{2} \equiv {{{{I_{2\; n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{I_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 4} \right)}}\theta}}} \\ {g_{3} \equiv {{{{- {I_{2\; n}\left( {\omega_{2}r} \right)}} \cdot \sin}\; 2\; n\; \theta} + {{{I_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 4} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},\mspace{14mu} {\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},\mspace{11mu} {\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv \sigma_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1483) \end{matrix}$

Converting eq. (1482) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 614} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{I_{{2\; n} + 1}\left( {\omega_{2}r} \right)} + {I_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{- {I_{{2\; n} + 1}\left( {\omega_{2}r} \right)}} + {I_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1484) \end{matrix}$

Converting eq. (1483) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {2\; {\mu \cdot {I_{{2n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 2} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{I_{2n}\left( {\omega_{2}r} \right)} + {I_{{2n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2n} + 2} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{- {I_{2n}\left( {\omega_{2}r} \right)}} + {I_{{2n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2n} + 2} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},{\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},{\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},{\sigma_{\theta}^{*} \equiv \sigma_{\theta}},{\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1485) \end{matrix}$

(7) Regarding K_(1SS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 615} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{- {K_{{2n} + 1}\left( {\omega_{1}r} \right)}} \cdot {\cos \left( {{2n} + 1} \right)}}\theta} + {{{K_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{K_{{2n} + 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 1} \right)}}\theta} + {{{K_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1486) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{{{K_{2n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2n\; \theta} - {{{K_{{2n} + 4}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 4} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{{{K_{2n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2n\; \theta} + {{{K_{{2n} + 4}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 4} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1487) \end{matrix}$

Converting eq. (1486) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 616} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{- {K_{{2n} + 1}\left( {\omega_{1}r} \right)}} + {K_{{2n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{K_{{2n} + 1}\left( {\omega_{1}r} \right)} + {K_{{2n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},{\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1488) \end{matrix}$

Converting eq. (1487) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{K_{2n}\left( {\omega_{1}r} \right)} - {K_{{2n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2n} + 2} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {{- G}\; \omega_{1}\left\{ {{K_{2n}\left( {\omega_{1}r} \right)} + {K_{{2n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2n} + 2} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},{\sigma_{\theta}^{*} \equiv \sigma_{\theta}},{\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1489) \end{matrix}$

(8) Regarding K_(2SS), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 617} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{K_{{2n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 1} \right)}}\theta} + {{{K_{{2n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{- {K_{{2n} + 1}\left( {\omega_{2}r} \right)}} \cdot {\sin \left( {{2n} + 1} \right)}}\theta} + {{{K_{{2n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},{\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1490) \\ \left\{ \begin{matrix} {g_{1} \equiv {{- 2}\; {\mu \cdot {K_{{2n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 2} \right)}}\theta}} \\ {g_{2} \equiv {{{{K_{2n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2n\; \theta} + {{{K_{{2n} + 4}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 4} \right)}}\theta}}} \\ {g_{3} \equiv {{{{K_{2n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2n\; \theta} - {{{K_{{2n} + 4}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 4} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},{\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},{\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},{\sigma_{y}^{*} \equiv \sigma_{y}},{\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1491) \end{matrix}$

Converting eq. (1490) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 618} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{K_{{2n} + 1}\left( {\omega_{2}r} \right)} + {K_{{2n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{- {K_{{2n} + 1}\left( {\omega_{2}r} \right)}} + {K_{{2n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},{\varphi_{\theta}^{*} \equiv \varphi_{74}}} \end{matrix} \right. & (1492) \end{matrix}$

Converting eq. (1491) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {{- 2}\; {\mu \cdot {K_{{2n} + 2}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 2} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{K_{2n}\left( {\omega_{2}r} \right)} + {K_{{2n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2n} + 2} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{K_{2n}\left( {\omega_{2}r} \right)} - {K_{{2n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2n} + 2} \right)}\theta}} \\ {{\sigma_{r} = {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},{\sigma_{\theta} = {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},{\tau_{r\; \theta} = {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},{\sigma_{\theta}^{*} \equiv \sigma_{\theta}},{\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1493) \end{matrix}$

[AA]

To the mode AA, the following eight sets in total belong: J_(1AA) of eq. (1253): J_(2AA) of eq. (1265): Y_(1AA) of eq. (1297): Y_(2AA) of eq. (1309): I_(1AA) of eq. (1341): I_(2AA), of eq. (1353): K_(1AA) of eq. (1385): and K_(2AA) of eq. (1397). As they serve as solution functions with respect to an arbitrary integer n, the total number is 8n.

(1) Regarding J_(1AA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 619} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{J_{{2n} + 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 1} \right)}}\theta} + {{{J_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{J_{{2n} + 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 1} \right)}}\theta} - {{{J_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1494) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{{{J_{2n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2n\; \theta} - {{{J_{{2n} + 4}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 4} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\left\{ {{{{J_{2n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2n\; \theta} + {{{J_{{2n} + 4}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 4} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1495) \end{matrix}$

Converting eq. (1494) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 620} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{J_{{2n} + 1}\left( {\omega_{1}r} \right)} + {J_{{2n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{J_{{2n} + 1}\left( {\omega_{1}r} \right)} - {J_{{2n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},{\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1496) \end{matrix}$

Converting eq. (1495) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{J_{2n}\left( {\omega_{1}r} \right)} - {J_{{2n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2n} + 2} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {G\; \omega_{1}\left\{ {{J_{2n}\left( {\omega_{1}r} \right)} + {J_{{2n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2n} + 2} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},{\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},{\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1497) \end{matrix}$

(2) Regarding J_(2AA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 621} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{J_{{2n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 1} \right)}}\theta} - {{{J_{{2n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{J_{{2n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 1} \right)}}\theta} + {{{J_{{2n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1498) \\ \left\{ \begin{matrix} {g_{1} \equiv {{- 2}\; {\mu \cdot {J_{{2n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 2} \right)}}\theta}} \\ {g_{2} \equiv {{{{J_{2n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2n\; \theta} + {{{J_{{2n} + 4}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 4} \right)}}\theta}}} \\ {g_{3} \equiv {{{{J_{2n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2n\; \theta} - {{{J_{{2n} + 4}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2n} + 4} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},{\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},{\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1499) \end{matrix}$

Converting eq. (1498) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 622} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{J_{{2n} + 1}\left( {\omega_{2}r} \right)} - {J_{{2n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{J_{{2n} + 1}\left( {\omega_{2}r} \right)} + {J_{{2n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},{\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1500) \end{matrix}$

Converting eq. (1499) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {{- 2}\; {\mu \cdot {J_{{2n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 2} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{J_{2n}\left( {\omega_{2}r} \right)} + {J_{{2n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2n} + 2} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{J_{2n}\left( {\omega_{2}r} \right)} - {J_{{2n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2n} + 2} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},{\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},{\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},{\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},{\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1501) \end{matrix}$

(3) Regarding Y_(1AA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 623} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{Y_{{2n} + 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 1} \right)}}\theta} + {{{Y_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{Y_{{2n} + 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 1} \right)}}\theta} - {{{Y_{{2n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},{\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1502) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{{{Y_{2n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2n\; \theta} - {{{Y_{{2n} + 4}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2n} + 4} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\left\{ {{{{Y_{2n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2n\; \theta} + {{{Y_{{2n} + 4}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2n} + 4} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},{\sigma_{y}^{*} \equiv {- \sigma_{y}}},{\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1503) \end{matrix}$

Converting eq. (1502) according to eqs. (969), (970) gives:

$\begin{matrix} \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{Y_{{2n} + 1}\left( {\omega_{1}r} \right)} + {Y_{{2n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{Y_{{2n} + 1}\left( {\omega_{1}r} \right)} - {Y_{{2n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},{\varphi_{\theta}^{*} \equiv {- ~\varphi_{\theta}}}} \end{matrix} \right. & (1504) \end{matrix}$

[Formula 624]

Converting eq. (1503) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{Y_{2\; n}\left( {\omega_{1}r} \right)} - {Y_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {G\; \omega_{1}\left\{ {{Y_{2\; n}\left( {\omega_{1}r} \right)} + {Y_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1505) \end{matrix}$

(4) Regarding Y_(2AA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 625} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{Y_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta} - {{{Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{Y_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta} + {{{Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv {- \varphi_{x}}},\mspace{14mu} {\varphi_{y}^{*} \equiv {- \varphi_{y}}}} \end{matrix} \right. & (1506) \\ \left\{ \begin{matrix} {g_{1} \equiv {{- 2}{\mu \cdot {Y_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}} \\ {g_{2} \equiv {{{{Y_{2\; n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{Y_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 4} \right)}}\theta}}} \\ {g_{3} \equiv {{{{Y_{2\; n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2\; n\; \theta} - {{{Y_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 4} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},{\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},{\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv {- \sigma_{x}}},\mspace{14mu} {\sigma_{y}^{*} \equiv {- \sigma_{y}}},\mspace{14mu} {\tau_{xy}^{*} \equiv {- \tau_{xy}}}} \end{matrix} \right. & (1507) \end{matrix}$

Converting eq. (1506) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 626} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{Y_{{2\; n} + 1}\left( {\omega_{2}r} \right)} - {Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{Y_{{2\; n} + 1}\left( {\omega_{2}r} \right)} + {Y_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv {- \varphi_{r}}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv {- \varphi_{\theta}}}} \end{matrix} \right. & (1508) \end{matrix}$

Converting eq. (1507) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {{- 2}{\mu \cdot {Y_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{Y_{2\; n}\left( {\omega_{2}r} \right)} + {Y_{{2\; n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{Y_{2\; n}\left( {\omega_{2}r} \right)} - {Y_{{2\; n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},{\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},{\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv {- \sigma_{r}}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv {- \sigma_{\theta}}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv {- \tau_{r\; \theta}}}} \end{matrix} \right. & (1509) \end{matrix}$

(5) Regarding I_(1AA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 627} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{I_{{2\; n} + 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta} - {{{I_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{I_{{2\; n} + 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta} + {{{I_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1510) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {G\; \omega_{1}\left\{ {{{{I_{2\; n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2\; n\; \theta} - {{{I_{{2\; n} + 4}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 4} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {G\; \omega_{1}\left\{ {{{{I_{2\; n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{I_{{2\; n} + 4}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 4} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv \sigma_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1511) \end{matrix}$

Converting eq. (1510) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 628} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{I_{{2\; n} + 1}\left( {\omega_{1}r} \right)} - {I_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{I_{{2\; n} + 1}\left( {\omega_{1}r} \right)} + {I_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1512) \end{matrix}$

Converting eq. (1511) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {G\; \omega_{1}\left\{ {{I_{2\; n}\left( {\omega_{1}r} \right)} - {I_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {G\; \omega_{1}\left\{ {{I_{2\; n}\left( {\omega_{1}r} \right)} + {I_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \sigma_{\theta}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1513) \end{matrix}$

(6) Regarding I_(2AA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 629} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{I_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta} + {{{I_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{I_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta} - {{{I_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1514) \\ \left\{ \begin{matrix} {g_{1} \equiv {2{\mu \cdot {I_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}} \\ {g_{2} \equiv {{{{I_{2\; n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{I_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 4} \right)}}\theta}}} \\ {g_{3} \equiv {{{{I_{2\; n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2\; n\; \theta} - {{{I_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 4} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},{\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},{\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv \sigma_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1515) \end{matrix}$

Converting eq. (1514) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 630} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{I_{{2\; n} + 1}\left( {\omega_{2}r} \right)} + {I_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{I_{{2\; n} + 1}\left( {\omega_{2}r} \right)} - {I_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1516) \end{matrix}$

Converting eq. (1515) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {2{\mu \cdot {I_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{I_{2\; n}\left( {\omega_{2}r} \right)} + {I_{{2\; n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{I_{2\; n}\left( {\omega_{2}r} \right)} - {I_{{2\; n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},{\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},{\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \sigma_{\theta}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1517) \end{matrix}$

(7) Regarding K_(1AA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 631} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{K_{{2\; n} + 1}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta} - {{{K_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{K_{{2\; n} + 1}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta} + {{{K_{{2\; n} + 3}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1518) \\ \left\{ \begin{matrix} {\sigma_{x} \equiv {{- G}\; \omega_{1}\left\{ {{{{K_{2\; n}\left( {\omega_{1}r} \right)} \cdot \sin}\; 2\; n\; \theta} - {{{K_{{2\; n} + 4}\left( {\omega_{1}r} \right)} \cdot {\sin \left( {{2\; n} + 4} \right)}}\theta}} \right\}}} \\ {\sigma_{y} \equiv {- \sigma_{x}}} \\ {\tau_{xy} \equiv {{- G}\; \omega_{1}\left\{ {{{{K_{2\; n}\left( {\omega_{1}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{K_{{2\; n} + 4}\left( {\omega_{1}r} \right)} \cdot {\cos \left( {{2\; n} + 4} \right)}}\theta}} \right\}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv \sigma_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1519) \end{matrix}$

Converting eq. (1518) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 632} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{K_{{2\; n} + 1}\left( {\omega_{1}r} \right)} - {K_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{K_{{2\; n} + 1}\left( {\omega_{1}r} \right)} + {K_{{2\; n} + 3}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1520) \end{matrix}$

Converting eq. (1519) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {\sigma_{r} \equiv {{- G}\; \omega_{1}\left\{ {{K_{2\; n}\left( {\omega_{1}r} \right)} - {K_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {\sigma_{\theta} \equiv {- \sigma_{r}}} \\ {\tau_{r\; \theta} \equiv {{- G}\; \omega_{1}\left\{ {{K_{2\; n}\left( {\omega_{1}r} \right)} + {K_{{2\; n} + 4}\left( {\omega_{1}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \sigma_{\theta}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1521) \end{matrix}$

(8) Regarding K_(2AA), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 633} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{x} \equiv {{{{K_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 1} \right)}}\theta} + {{{K_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 3} \right)}}\theta}}} \\ {\varphi_{y} \equiv {{{{K_{{2\; n} + 1}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 1} \right)}}\theta} - {{{K_{{2\; n} + 3}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 3} \right)}}\theta}}} \\ {{\varphi_{x}^{*} \equiv \varphi_{x}},\mspace{14mu} {\varphi_{y}^{*} \equiv \varphi_{y}}} \end{matrix} \right. & (1522) \\ \left\{ \begin{matrix} {g_{1} \equiv {2{\mu \cdot {K_{{2\; n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 2} \right)}}\theta}} \\ {g_{2} \equiv {{{{K_{2\; n}\left( {\omega_{2}r} \right)} \cdot \sin}\; 2\; n\; \theta} + {{{K_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2\; n} + 4} \right)}}\theta}}} \\ {g_{3} \equiv {{{{K_{2\; n}\left( {\omega_{2}r} \right)} \cdot \cos}\; 2\; n\; \theta} + {{{K_{{2\; n} + 4}\left( {\omega_{2}r} \right)} \cdot {\cos \left( {{2\; n} + 4} \right)}}\theta}}} \\ {{\sigma_{x} \equiv {G\; {\omega_{2}\left( {g_{1} - g_{2}} \right)}}},{\sigma_{y} \equiv {G\; {\omega_{2}\left( {g_{1} + g_{2}} \right)}}},{\tau_{xy} \equiv {G\; \omega_{2}g_{3}}}} \\ {{\sigma_{x}^{*} \equiv \sigma_{x}},\mspace{14mu} {\sigma_{y}^{*} \equiv \sigma_{y}},\mspace{14mu} {\tau_{xy}^{*} \equiv \tau_{xy}}} \end{matrix} \right. & (1523) \end{matrix}$

Converting eq. (1522) according to eqs. (969), (970) gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 634} \right\rbrack & \; \\ \left\{ \begin{matrix} {\varphi_{r} \equiv {\left\{ {{K_{{2\; n} + 1}\left( {\omega_{2}r} \right)} + {K_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2\; n} + 2} \right)}\theta}} \\ {\varphi_{\theta} \equiv {\left\{ {{K_{{2\; n} + 1}\left( {\omega_{2}r} \right)} - {K_{{2\; n} + 3}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2\; n} + 2} \right)}\theta}} \\ {{\varphi_{r}^{*} \equiv \varphi_{r}},\mspace{14mu} {\varphi_{\theta}^{*} \equiv \varphi_{\theta}}} \end{matrix} \right. & (1524) \end{matrix}$

Converting eq. (1523) according to eqs. (977), (978), (979) gives:

$\begin{matrix} \left\{ \begin{matrix} {h_{1} \equiv {{- 2}\; {\mu \cdot {K_{{2n} + 2}\left( {\omega_{2}r} \right)} \cdot {\sin \left( {{2n} + 2} \right)}}\theta}} \\ {h_{2} \equiv {\left\{ {{K_{2n}\left( {\omega_{2}r} \right)} + {K_{{2n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\sin \left( {{2n} + 2} \right)}\theta}} \\ {h_{3} \equiv {\left\{ {{- {K_{2n}\left( {\omega_{2}r} \right)}} + {K_{{2n} + 4}\left( {\omega_{2}r} \right)}} \right\} {\cos \left( {{2n} + 2} \right)}\theta}} \\ {{\sigma_{r} \equiv {G\; {\omega_{2}\left( {h_{1} - h_{2}} \right)}}},{\sigma_{\theta} \equiv {G\; {\omega_{2}\left( {h_{1} + h_{2}} \right)}}},{\tau_{r\; \theta} \equiv {G\; \omega_{2}h_{3}}}} \\ {{\sigma_{r}^{*} \equiv \sigma_{r}},\mspace{14mu} {\sigma_{\theta}^{*} \equiv \sigma_{\theta}},\mspace{14mu} {\tau_{r\; \theta}^{*} \equiv \tau_{r\; \theta}}} \end{matrix} \right. & (1525) \end{matrix}$

11.4.26 Boundary Condition and Adjoint Boundary Condition, as Well as Eigenfunction Set

For comparison with analytical solutions in Section 11.4.10, a problem is solved with a boundary condition given in Section 11.4.9. More specifically, it is determined, regarding a ring on which uniform gravity is acting in the y direction, how the outer edge thereof should be supported so that a boundary condition for the inner edge thereof is such that a surface force is zero and a displacement is zero. This is a problem that has been considered insoluble conventionally.

A boundary condition imposed on an eigenfunction is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 635} \right\rbrack & \; \\ {\left. {{{\varphi_{r}}_{r = {\gamma \; R}} = 0}{\varphi_{{\theta }_{r = {\gamma \; R}}} = 0}\begin{matrix} {{\sigma_{Er}}_{r = {\gamma \; R}} = 0} \\ {\tau_{{{{Er}\; \theta}}_{r = {\gamma \; R}}} = 0} \end{matrix}} \right\} \left\{ \begin{matrix} {{p_{Er}}_{r = {\gamma \; R}} = 0} \\ {p_{{{E\; \theta}}_{r = {\gamma \; R}}} = 0} \end{matrix} \right.} & (1526) \end{matrix}$

The index E represents a stress or a surface force generated by the eigenfunction, as described in Section 4.2. Eq. (1526) are equal to the condition equations (1018), (1019), and (1069). In the same manner as that for obtaining eq. (1085), the boundary term R_(E) of eq. (91) is expressed as follows with polar coordinate components:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 636} \right\rbrack} & \; \\ {R_{E} \equiv {{\frac{1}{G}{\int_{C}{\left( {{p_{Er} \cdot \sigma_{r}^{*}} + {p_{E\; \theta} \cdot \varphi_{\theta}^{*}}} \right)\ {c}}}} - {\frac{1}{G}{\int_{C}{\left( {{\varphi_{r} \cdot p_{Er}^{*}} + {\varphi_{\theta} \cdot p_{E\; \theta}^{*}}} \right){c}}}}}} & (1527) \end{matrix}$

Therefore, the adjoint boundary condition is given as:

$\begin{matrix} {\left. {{{{{{\varphi_{r}^{*}}_{r = \; R} = 0}\varphi_{\theta}^{*}}}_{r = R} = 0}\begin{matrix} {{\sigma_{Er}^{*}}_{r = R} = 0} \\ {{\tau_{{Er}\; \theta}^{*}}_{r = R} = 0} \end{matrix}} \right\} \left\{ \begin{matrix} {{p_{Er}^{*}}_{r = R} = 0} \\ {{p_{E\; \theta}^{*}}_{r = R} = 0} \end{matrix} \right.} & (1528) \end{matrix}$

This is equal to eq. (1070), which is a condition such that on the outer edge (r=R), the surface force is zero and the displacement is zero. As the condition of eq. (1526) and the condition of eq. (1528) are different, this is a non-self-adjoint boundary condition.

Only the function set of the mode AS is able to express the solution in the case of uniform gravity in the y direction, and the other three modes, SA, SS, AA do not contribute to this. To the mode AS, the following eight sets in total belong: J_(1AS) of eq. (1251); J_(2AS) of eq. (1263); Y_(2AS) of eq. (1295); Y_(2AS) of eq. (1307); I_(1AS) of eq. (1339); I_(2AS) of eq. (1351); K_(1AS) of eq. (1383); and K_(2AS) of eq. (1395). Besides, they serve as solution functions with respect to an arbitrary integer n, the total number is 8n. However, only one function set of n=0 among those of the mode AS is able to expressing the solution in the case of uniform gravity, and the sets of n≠0 do not contribute to this.

The primal eigenfunction and the dual eigenfunction, as well as stress functions based on these are collectively given as F. Then, J_(1AS) of eq. (1251) is multiplied by a coefficient c_(J1), J_(2AS) of eq. (1263) is multiplied by a coefficient c_(J2), Y_(1AS) of eq. (1295) is multiplied by a coefficient c_(Y1), Y_(2AS) of eq. (1307) is multiplied by a coefficient c_(Y2), I_(1AS) of eq. (1339) is multiplied by a coefficient c_(I1), I_(2AS) of eq. (1351) is multiplied by a coefficient c_(I2), K_(1AS) of eq. (1383) is multiplied by a coefficient c_(K1), and K_(2AS) of eq. (1395) is multiplied by a coefficient c_(K2), and they are added and equated to F. In other words, the following is given:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 637} \right\rbrack & \; \\ {F \equiv \left\{ {\begin{matrix} {\varphi_{x},\varphi_{y}} \\ {\sigma_{x},\sigma_{y},\tau_{xy}} \\ {\varphi_{x}^{*},\varphi_{y}^{*}} \\ {\sigma_{x}^{*},\sigma_{y}^{*},\tau_{xy}^{*}} \end{matrix},{\underset{8 \times 1}{\left\{ c_{F} \right\}} \equiv \begin{Bmatrix} c_{J\; 1} \\ c_{J\; 2} \\ c_{Y\; 1} \\ c_{Y\; 2} \\ c_{I\; 1} \\ c_{I\; 2} \\ c_{K\; 1} \\ c_{K\; 2} \end{Bmatrix}}} \right.} & (1529) \end{matrix}$

Then, the following equation is created:

F=c _(J1) ·J _(1AS) +c _(J2) ·J _(2AS) +c _(Y1) ·Y _(1AS) +c _(Y2) ·Y _(2AS) +c ₁₁ ·I _(1AS) +c ₁₂ ·I _(2AS) +c _(K1) ·K _(1AS) +c _(K2) ·K _(2AS)  (1530)

The coefficients c_(F) are determined so that F satisfies the imposed boundary condition. It is convenient to give the boundary condition with polar coordinate components as is the case with eqs. (1526), (1528). Then we use following expressions, J_(1AS) with eqs. (1432) and (1433), J_(2AS) with eqs. (1436) and (1437), Y_(1AS) with eqs. (1440) and (1441), Y_(2AS) with eqs. (1444) and (1445), I_(1AS) with eqs. (1448) and (1449), I_(2AS) with eqs. (1452) and (1453), K_(1AS) with eqs. (1456) and (1457), and K_(2AS) with eqs. (1460) and (1461). As the boundary conditions are eight in total, and the coefficients c_(F) are eight in total as well, the following simultaneous equation is obtained:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 638} \right\rbrack & \; \\ {{\underset{8 \times 8}{\lbrack A\rbrack}\underset{8 \times 1}{\left\{ c_{F} \right\}}} = \underset{8 \times 1}{\left\{ 0 \right\}}} & (1531) \end{matrix}$

Components of the matrix A are given as:

$\begin{matrix} {\underset{8 \times 8}{\lbrack A\rbrack} = \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} & a_{17} & a_{18} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} & a_{27} & a_{28} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} & a_{37} & a_{38} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} & a_{47} & a_{48} \\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56} & a_{57} & a_{58} \\ a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66} & a_{67} & a_{68} \\ a_{71} & a_{72} & a_{73} & a_{74} & a_{75} & a_{76} & a_{77} & a_{78} \\ a_{81} & a_{82} & a_{83} & a_{84} & a_{85} & a_{86} & a_{87} & a_{88} \end{bmatrix}} & (1532) \end{matrix}$

[Formula 639]

Eigenvalues ω₁,ω₂ of eqs. (1103), (1104) are given with χ₁,χ₂ as follows:

$\begin{matrix} \left. {\omega_{1} \equiv {\frac{1}{R}\sqrt{\lambda}} \equiv {\frac{1}{R}\chi_{1}}}\Rightarrow{\chi_{1} \equiv \sqrt{\lambda}} \right. & (1533) \\ \left. {\omega_{2} \equiv {\frac{1}{R}\sqrt{\frac{\lambda}{1 + \mu}}} \equiv {\frac{1}{R}\chi_{2}}}\Rightarrow{\chi_{2} \equiv \sqrt{\frac{\lambda}{1 + \mu}}} \right. & (1534) \end{matrix}$

Then, the equations are rearranged, and the components of the matrix A can be simplified as follows. Here, let us note that not only the function set of n=0 but also the function sets of n≠0 satisfy the boundary condition. In the case of the y-direction uniform gravity load, the function set of n=0 expresses a solution, but in the other load states, the function sets of n≠0 can express a solution. Therefore, the components of the matrix A are expressed with use of n. The first and second columns of the matrix A are given as:

[Formula 640]

c ₁ ≡J _(2n)(χ₁), c ₅ ≡J _(2n)(χ₂)

c ₂ ≡J _(2n+2)(χ₁), c ₆ ≡J _(2n+2)(χ₂)

c ₃ ≡J _(2n)(γχ₁), c ₇ ≡J _(2n)(γχ₂)

c ₄ ≡J _(2n+2)(γχ₁), c ₈ ≡J _(2n+2)(γχ₂)  (1535)

Then, we obtain:

[Formula 641]

a ₁₁ ≡c ₁ , a ₁₂ ≡c ₅

a ₂₁ ≡c ₂ , a ₂₂ ≡−c ₆

a ₃₁ ≡c ₁ +c ₂ , a ₃₂≡0

a ₄₁≡0, a ₄₂ ≡c ₅ +c ₆

a ₅₁ ≡c ₃ , a ₅₂ ≡c ₇

a ₆₁ ≡c ₄ , a ₆₂ ≡−c ₈

a ₇₁ ≡c ₃ +c ₄ , a ₇₂≡0

a ₈₁≡0, a ₈₂ ≡c ₇ +c ₈  (1536)

The third and fourth columns of the matrix A are given as:

[Formula 642]

c ₉ ≡Y _(2n)(χ₁), c ₁₃ ≡Y _(2n)(χ₂)

c ₁₀ ≡Y _(2n+2)(χ₁), c ₁₄ ≡Y _(2n+2)(χ₂)

c ₁₁ ≡Y _(2n)(γχ₁), c ₁₅ ≡Y _(2n)(χ₂)

c ₁₂ ≡Y _(2n+2)(γχ₁), c ₁₆ ≡Y _(2n+2)(χ₂)  (1537)

Then, we obtain:

[Formula 643]

a ₁₃ ≡c ₉ , a ₁₄ ≡c ₁₃

a ₂₃ ≡c ₁₀ , a ₂₄ ≡−c ₁₄

a ₃₃ ≡c ₉ +c ₁₀ , a ₃₄≡0

a ₄₃≡0, a ₄₄ ≡c ₁₃ +c ₁₄

a ₅₃ ≡c ₁₁ , a ₅₄ ≡c ₁₅

a ₆₃ ≡c ₁₂ , a ₆₄ ≡−c ₁₆

a ₇₃ ≡c ₁₁ +c ₁₂ , a ₇₄≡0

a ₈₃≡0, a ₈₄ ≡c ₁₅ +c ₁₆  (1538)

The fifth and sixth columns of the matrix A are given as:

[Formula 644]

c ₁₇ ≡I _(2n)(χ₁), c ₂₁ ≡I _(2n)(χ₂)

c ₁₈ ≡I _(2n+2)(χ₁), c ₂₂ ≡I _(2n+2)(χ₂)

c ₁₉ ≡I _(2n)(γχ₁), c ₂₃ ≡I _(2n)(γχ₂)

c ₂₀ ≡I _(2n+2)(γχ₁), c ₂₄ ≡I _(2n+2)(γχ₂)  (1539)

Then, we obtain:

[Formula 645]

a ₁₅ ≡−c ₁₇ , a ₁₆ ≡−c ₂₁

a ₂₅ ≡c ₁₈ , a ₂₆ ≡−c ₂₂

a ₃₅ ≡c ₁₇ −c ₁₈ , a ₃₆≡0

a ₄₅≡0, a ₄₆ ≡c ₂₁ −c ₂₂

a ₅₅ ≡c ₁₉ , a ₅₆ ≡c ₂₃

a ₆₅ ≡−c ₂₀ , a ₆₆ ≡c ₂₄

a ₇₅ ≡c ₁₉ +c ₂₀ , a ₇₆≡0

a ₈₅≡0, a ₈₆ ≡−c ₂₃ +c ₂₄  (1540)

The seventh and eighth columns of the matrix A are given as:

[Formula 646]

c ₂₅ ≡K _(2n)(χ₁), c ₂₉ ≡K _(2n)(χ₂)

c ₂₆ ≡K _(2n+2)(χ₁), c ₃₀ ≡K _(n+2)(χ₂)

c ₂₇ ≡K _(2n)(γχ₁), c ₃₁ ≡K _(2n)(γχ₂)

c ₂₈ ≡K _(2n+2)(γχ₁), c ₃₂ ≡K _(2n+2)(γχ₂)  (1541)

Then, we obtain:

[Formula 647]

a ₁₇ ≡−c ₂₅ , a ₁₈ ≡−c ₂₉

a ₂₇ ≡c ₂₆ , a ₂₈ ≡−c ₃₀

a ₃₇ ≡c ₂₅ −c ₂₆ , a ₃₈≡0

a ₄₇≡0, a ₄₈ ≡c ₂₉ −c ₃₀

a ₅₇ ≡c ₂₇ , a ₅₈ ≡c ₃₁

a ₆₇ ≡−c ₂₈ , a ₆₈ ≡c ₃₂

a ₇₇ ≡−c ₂₇ +c ₂₈ , a ₇₈≡0

a ₈₇≡0, a ₈₈ ≡−c ₃₁ +c ₃₂  (1542)

The determinant of the matrix A become functions of χ₁, χ₂, and consequently functions of the eigenvalue λ according to eqs. (1533), (1534). By solving an characteristic equation that makes the determinant zero, we obtain the eigenvalue λ. Further, according to eq. (1531), we obtain an eigenvector c_(F) corresponding to the eigenvalue λ. As the characteristic equation and the eigenvector have great length, it is impossible to describe the same herein unfortunately. However, the result of solution can be described numerically.

11.4.27 Primal Eigenfunction and Dual Eigenfunction

In a plane stress state, the material constants are given as follows, as is the case with the analytical solution described in Section 11.4.10:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 648} \right\rbrack & \; \\ {{v \equiv \frac{3}{10}},{{\mu \equiv \frac{1 + v}{1 - v}} = \frac{13}{7}}} & \begin{matrix} {\mspace{115mu} (1026)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

The inner diameter ratio γ is given as:

$\begin{matrix} {\gamma \equiv \frac{3}{10}} & \begin{matrix} {\mspace{115mu} (1027)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

Solving the characteristic equation obtained from eq. (1531) in which n=0 gives four lower-order eigenvalues λ in the vicinities of:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 649} \right\rbrack & \; \\ \left\{ \begin{matrix} {5.317 \times 10^{0}} \\ {1.882 \times 10^{1}} \\ {5.080 \times 10^{1}} \\ {1.157 \times 10^{2}} \end{matrix} \right. & (1543) \end{matrix}$

When eigenvectors c^(F) corresponding to the eigenvalues λ are determined and substituted into eq. (1530), F is settled. F includes a primal eigenfunction, a dual eigenfunction, and a stress function based on the same.

States of deformation and stress distribution in the No. 1 mode (λ=5.317×10⁰) are shown in FIG. 47. The upper part shows states determined according to the primal eigenfunction, wherein the surface force is zero and the displacement is zero also on the inner edge. The lower part shows states determined according to the dual eigenfunction, wherein the surface force is zero and the displacement is zero on the outer edge. The states of deformation in the upper part appear similar to deformation determined according to the analytical solution shown in FIG. 42, but the stress distribution thereof is significantly different. For example, in the case of the analytical solution, stress distribution is seen regarding σ_(r), while in the case of the primal eigenfunction, similar stress distribution is seen regarding σ_(θ).

States of deformation and stress distribution in No. 2 mode (λ=1.882×10¹) are shown in FIG. 48. The upper part shows states determined according to the primal eigenfunction, wherein the surface force is zero and the displacement is zero also on the inner edge. The lower part shows states determined according to the dual eigenfunction, wherein the surface force is zero and the displacement is zero on the outer edge. Focusing on σ_(θ) according to the primal eigenfunction, we can see that the distribution is similar to that of σ_(θ) in the No. 1 mode. Further, we can see that the distribution of σ_(r) is similar to that according to the analytical solution. Therefore, from the difference between No. 1 mode and No. 2 mode, it can be expected to obtain states similar to those according to the analytical solution.

States of deformation and stress distribution in No. 3 mode (λ=5.080×10¹) are shown in FIG. 49. The states of deformation are slightly complicated.

States of deformation and stress distribution in No. 4 mode (λ=1.157×10²) are shown in FIG. 50. The states of deformation are further complicated.

11.4.28 Eigenfunction Method

A solution is determined by the eigenfunction method described in Section 5.1. As is the case with the analytical solution described in Section 11.4.10, load coefficients c_(x), c_(y) are given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 650} \right\rbrack & \; \\ {\begin{Bmatrix} c_{x} \\ c_{y} \end{Bmatrix} \equiv \begin{Bmatrix} 0 \\ 1 \end{Bmatrix}} & \begin{matrix} {\mspace{115mu} (1028)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

Then, a displacement is made dimensionless with the outer radius R.

Results of calculation performed by using up to No. 2 mode are shown in FIG. 51. The stress σ_(θ) is cancelled, and we find that the distribution of σ_(r) is similar to the analytical solution.

Results of calculation performed by using up to No. 3 mode are shown in FIG. 52. We find that the distribution of the stress σ_(r) is closer to that of the analytical solution.

Results of calculation performed by using up to No. 10 mode are shown in FIG. 53. We find that the stress distribution is closer to that of the analytical solution.

Results of calculation performed by using up to No. 30 mode are shown in FIG. 54. We find that the results have substantially no difference from the results of calculation performed by using up to No. 10 mode, and the stress distribution and the state of deformation are generally closer to the analytical solution.

11.5 Source and Vortex

This section should better be included in Section 10.

11.5.1 Differential Equation

The problem of two-dimensional irrotational flow is described by simultaneous partial differential equations of the following equation of continuity (18) and the following equation of vortex-free condition (19) in the orthogonal coordinate system:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 651} \right\rbrack & \; \\ {{\frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y}} = 0} & \begin{matrix} {\mspace{146mu} (18)} \\ ({Aforementioned}\;) \end{matrix} \\ {{\frac{\partial u_{y}}{\partial x} - \frac{\partial u_{x}}{\partial y}} = 0} & \begin{matrix} {\mspace{140mu} (19)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

where u_(x),u_(y), represent velocities in the x,y directions, respectively. If the velocity potential φ is defined as satisfying the following relationship with velocity of flow

$\begin{matrix} {{u_{x} \equiv \frac{\partial\varphi}{\partial x}},{u_{y} \equiv \frac{\partial\varphi}{\partial y}},} & (1544) \end{matrix}$

and when this equation is substituted into eq. (19), the equal sign is established always. Therefore, only using the velocity potential satisfies the vortex-free condition equation (19). Further, substituting eq. (1544) into eq. (18), we obtain the following Laplace's equation (Laplace's equation):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 652} \right\rbrack & \; \\ {{{\nabla^{2}\varphi} \equiv {\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)\varphi}} = 0} & (1545) \end{matrix}$

which represents the equation of continuity. Therefore, the problem of irrotational flow is called a two-dimensional potential flow.

On the other hand, if the flow function ψ is defined as satisfying the following relationship with the velocity of flow

$\begin{matrix} {{u_{x} \equiv \frac{\partial\psi}{\partial y}},\mspace{14mu} {u_{y} \equiv \frac{\partial\psi}{\partial x}},} & (1546) \end{matrix}$

and when this equation is substituted into eq. (18), the equal sign is established always. Therefore, only using the flow function satisfies the equation of continuity (18). Further, substituting eq. (1546) into eq. (19), we obtain the following Laplace's equation:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 653} \right\rbrack & \; \\ {{{- {\nabla^{2}\psi}} \equiv {{- \left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)}\psi}} = 0} & (1547) \end{matrix}$

which represents the vortex-free condition.

In the case where a source is represented carefully, the right sides of eqs. (18) and (1545) become Dirac's δ functions (Dirac's delta functions). Similarly, in the case where a vortex is represented carefully, the right sides of eqs. (19) and (1547) become Dirac's δ functions. To eliminate the necessity of switching the equations according to the characteristics of the source or the vortex, eq. (18) may be taken as

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 654} \right\rbrack & \; \\ {{\frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y}} = f_{x}} & (1548) \end{matrix}$

and f_(x) is recognized as as source distribution, and similarly, eq. (19) may be given as

$\begin{matrix} {{{- \frac{\partial u_{x}}{\partial y}} + \frac{\partial u_{y}}{\partial x}} = f_{y}} & (1549) \end{matrix}$

and f_(y) may be recognized as vortex distribution. Further, if the problem is dealt with by the simultaneous differential equations (1548), (1549), the boundary function can be designated with the velocities u_(x), u_(y), which gives us advantages.

Velocities u_(x), u_(y) determined as primal solutions are referred to as primal velocities, and written as u₁, u₂. If the right-hand sides of the equations are written as f₁, f₂, then the simultaneous partial differential equations (1548), (1549) are transformed to the form of eq. (23):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 655} \right\rbrack & \; \\ {{\sum\limits_{j}\; {L_{ij}u_{j}}} = f_{i}} & \underset{({Aforementioned})}{(23)} \end{matrix}$

This equation is a basis for application of the eigenfunction method to a problem of hydrodynamics.

11.5.2 Polar Coordinate Solution of Harmonic Equation

In Section 11.4.8, solutions of biharmonic equations are obtained, some of which satisfy a harmonic equation (Laplace's equation). Since the polar coordinate format of the differential operator is given by eq. (992),

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 656} \right\rbrack & \; \\ {{{{\nabla^{2}\varphi} \equiv {\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)\varphi}} = {\left( {\frac{\partial^{2}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}}} \right)\varphi}},} & \underset{({Aforementioned})}{(992)} \end{matrix}$

eq. (1545) is transformed to:

$\begin{matrix} {{{\nabla^{2}\varphi} \equiv {\left( {\frac{\partial^{2}}{\partial r^{2}} + {\frac{1}{r}\frac{\partial}{\partial r}} + {\frac{1}{r^{2}}\frac{\partial^{2}}{\partial\theta^{2}}}} \right)\varphi}} = 0.} & (1550) \end{matrix}$

A solution satisfying this is transformed into a variable separation form as follows:

φ(r,θ)≡A(r)·B(θ)  (1551)

Then, eq. (1550) is transformed to:

$\begin{matrix} {{{r^{2}\frac{\overset{¨}{A}(r)}{A(r)}} + {r\frac{\overset{.}{A}(r)}{A(r)}}} = {{- \frac{\overset{¨}{B}(\theta)}{B(\theta)}} \equiv v^{2}}} & (1552) \end{matrix}$

where v represents an arbitrary constant.

A differential equation obtained from eq. (1552) is given as:

[Formula 657]

B ^(&&)(θ)+v ² B(θ)=0  (1553)

And a solution of this equation is given as:

B(θ)=sin vθ, cos vθ  (1554)

Similarly, a differential equation obtained from eq. (1552) is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 658} \right\rbrack & \; \\ {{{\overset{¨}{A}(r)} + {\frac{1}{r}{\overset{.}{A}(r)}} - {\left( \frac{v}{r} \right)^{2}{A(r)}}} = 0} & (1555) \end{matrix}$

And a solution of this equation is given as:

A(r)=r ^(v) ,r ^(−v)  (1556)

Consequently, we obtain the following solution:

$\begin{matrix} {{{\varphi \left( {r,\theta} \right)} \equiv {{A(r)} \cdot {B(\theta)}}} = {\begin{pmatrix} r^{v} \\ r^{- v} \end{pmatrix} \times \begin{pmatrix} {\sin \; v\; \theta} \\ {\cos \; v\; \theta} \end{pmatrix}}} & (1557) \end{matrix}$

As a derived function of a velocity potential represents a velocity, it is necessary to obtain a solution of periodicity in the θ direction. Therefore, let v be an integer n, and we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 659} \right\rbrack & \; \\ {{{\varphi \left( {r,\theta} \right)} \equiv {{A(r)} \cdot {B(\theta)}}} = {\begin{pmatrix} r^{n} \\ r^{- n} \end{pmatrix} \times \begin{pmatrix} {\sin \; n\; \theta} \\ {\cos \; n\; \theta} \end{pmatrix}}} & (1558) \end{matrix}$

In a special case where v=0 in eq. (1552), a solution given as eq. (1560) is obtained as a solution of a differential equation (1559):

[Formula 660]

B ^(&&)(θ)=0  (1559)

B(θ)=1,θ  (1560)

Similarly, a solution of a differential equation (1561) is given as eq. (1562):

$\begin{matrix} {{{\overset{¨}{A}(r)} + {\frac{1}{r}{\overset{.}{A}(r)}}} = 0} & (1561) \\ {{{A(r)} = 1},{\ln \; r}} & (1562) \end{matrix}$

Consequently, we obtain the following solution:

$\begin{matrix} {{{\varphi \left( {r,\theta} \right)} \equiv {{A(r)} \cdot {B(\theta)}}} = {\begin{pmatrix} 1 \\ {\ln \; r} \end{pmatrix} \times \begin{pmatrix} 1 \\ \theta \end{pmatrix}}} & (1563) \end{matrix}$

As a derived function of a velocity potential represents a velocity, it is necessary to obtain a solution of periodicity in the θ direction, and therefore, θ·ln r is inappropriate as a solution. Further, 1.1 represents nothing but a velocity of zero. Therefore, the following two are solutions:

[Formula 661]

φ(r,θ)≡ln r,θ  (1564)

The first solution ln r of this equation represents a source, and the second solution θ represents a vortex.

11.5.3 Orthogonal Coordinate Solution of Harmonic Equation

When a solution satisfying the following harmonic equation (Laplace's equation) (1545) is given in a variable separation form given as eq. (1565)

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 662} \right\rbrack & \; \\ {{{\nabla^{2}\varphi} \equiv {\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}} \right)\varphi}} = 0} & \underset{({Aforementioned})}{(1545)} \\ {{{\varphi \left( {x,y} \right)} \equiv {{X(x)} \cdot {Y(y)}}},} & (1565) \end{matrix}$

eq. (1545) is transformed to

$\begin{matrix} {\frac{\overset{¨}{X}(r)}{X(r)} = {{- \frac{\overset{¨}{Y}(y)}{Y(y)}} \equiv v^{2}}} & (1566) \end{matrix}$

where v represents an arbitrary constant.

A differential equation obtained from eq. (1556) is given as:

[Formula 663]

X ^(&&)(x)−v ² X(x)=0  (1567)

And a solution of this equation is given as:

X(x)=sin hvx,cos hvx  (1568)

Similarly, a differential equation obtained from eq. (1566) is given as:

Y ^(&&)(y)+v ² Y(y)=0  (1569)

And a solution of this equation is given as:

Y(y)=sin vy,cos vy  (1570)

Consequently, we obtain the following solution:

$\begin{matrix} {{{\varphi \left( {x,y} \right)} \equiv {{X(x)} \cdot {Y(y)}}} = {\begin{pmatrix} {\sinh \mspace{14mu} {vx}} \\ {\cosh \mspace{14mu} {vx}} \end{pmatrix} \times \begin{pmatrix} {\sin \mspace{14mu} {vy}} \\ {\cos \mspace{14mu} {vy}} \end{pmatrix}}} & (1571) \end{matrix}$

Understanding that inverting the sign of the constant part of eq. (1566) gives

$\begin{matrix} {{{- \frac{X^{\&\&}(x)}{X(x)}} = {\frac{Y^{\&\&}(y)}{Y(y)} \equiv v^{2}}},} & (1572) \end{matrix}$

a differential equation obtained from eq. (1572) is given as:

[Formula 664]

X ^(&&)(x)+v ² X(x)=0  (1573)

And a solution thereof is given as:

X(x)=sin vx,cos vx  (1574)

Similarly, a differential equation obtained from eq. (1572) is given as:

Y ^(&&)(y)−v ² Y(y)=0  (1575)

And a solution thereof is given as:

Y(y)=sin hvy,cos hvy  (1576)

Consequently, we obtain the following solution:

$\begin{matrix} {{{\varphi \left( {x,t} \right)} \equiv {{X(x)} \cdot {Y(y)}}} = {\begin{pmatrix} {\sin \mspace{14mu} {vx}} \\ {\cos \mspace{14mu} {vx}} \end{pmatrix} \times \begin{pmatrix} {\sinh \mspace{14mu} {vy}} \\ {\cosh \mspace{14mu} {vy}} \end{pmatrix}}} & (1577) \end{matrix}$

In a special case where v=0 in eq. (1572), a solution given as eq. (1579) is obtained as a solution of a differential equation (1578):

[Formula 665]

X ^(&&)(x)=0  (1578)

X(x)=1,x  (1579)

Similarly, a solution of a differential equation (1580) is given as eq. (1581):

Y ^(&&)(y)=0  (1580)

Y(y)=1,y  (1581)

Consequently, we obtain the following solution:

$\begin{matrix} {{{\varphi \left( {x,y} \right)} \equiv {{X(x)} \cdot {Y(y)}}} = {\begin{pmatrix} 1 \\ x \end{pmatrix} \times \begin{pmatrix} 1 \\ y \end{pmatrix}}} & (1582) \end{matrix}$

11.5.4 Orthogonal Coordinate Solution of Harmonic Characteristic Equation

When a solution satisfying the following harmonic characteristic equation (1583) is given in a variable separation form given as eq. (1584)

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 666} \right\rbrack & \; \\ {{{\left( {\nabla^{2}{+ \omega^{2}}} \right)\varphi} \equiv {\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \omega^{2}} \right)\varphi}} = 0} & (1583) \\ {{{\varphi \left( {x,y} \right)} \equiv {{X(x)} \cdot {Y(y)}}},} & (1584) \end{matrix}$

eq. (1583) is transformed to

$\begin{matrix} {{\frac{X^{\&\&}(x)}{X(x)} + \frac{Y^{\&\&}(y)}{Y(y)} + \omega^{2}} = 0.} & (1585) \end{matrix}$

Assuming that solutions having periodicity are obtained in both of the x and y directions, and giving

$\begin{matrix} {\frac{X^{\&\&}(x)}{X(x)} \equiv {- \alpha^{2}}} & (1586) \\ {{\frac{Y^{\&\&}(y)}{Y(y)} \equiv {- \beta^{2}}},} & (1587) \end{matrix}$

then, we obtain:

[Formula 667]

α²+β²=ω²  (1588)

A differential equation obtained from eq. (1586) is given as:

X ^(&&)(x)+α² X(x)=0  (1589)

And a solution of this equation is given as:

X(x)=sin φx,cos φx  (1590)

Similarly, a differential equation obtained from eq. (1587) is given as:

Y ^(&&)(y)+β² Y(y)=0  (1591)

And a solution of this equation is given as:

Y(y)=sin βy,cos βy  (1592)

Consequently, we obtain the following solution:

$\begin{matrix} {{{\varphi \left( {x,y} \right)} \equiv {{X(x)} \cdot {Y(y)}}} = {\begin{pmatrix} {\sin \mspace{14mu} \alpha \; x} \\ {\cos \mspace{14mu} \alpha \; x} \end{pmatrix} \times \begin{pmatrix} {\sin \mspace{14mu} \beta \; y} \\ {\cos \mspace{14mu} \beta \; y} \end{pmatrix}}} & (1593) \end{matrix}$

11.5.5 Solution of Source

Let the intensity of source at the origin of coordinates be Q[m²/s], and the velocity potential φ is given as follows according to eq. (1564):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 668} \right\rbrack & \; \\ {\varphi = {{\frac{Q}{2\pi}\ln \; r} = {\frac{Q}{2\pi}\ln \sqrt{x^{2} + y^{2}}}}} & (1584) \end{matrix}$

Differentiating this gives velocities of u_(x),u_(y), which are given as:

$\begin{matrix} {{{u_{x} \equiv \frac{\partial\varphi}{\partial x}} = {\frac{Q}{2\pi}\frac{x}{x^{2} + y^{2}}}}{{u_{y} \equiv \frac{\partial\varphi}{\partial y}} = {\frac{Q}{2\pi}\frac{y}{x^{2} + y^{2}}}}} & (1595) \end{matrix}$

In the case where the intensity Q=1, the source is called a source of unit intensity. Let the velocity vector by the components u_(x),u_(y) of eq. (1595) be v, and we obtain:

$\begin{matrix} {{{div}(v)} = {{\frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y}} = 0}} & (1596) \\ {{{rot}(v)} = {{\frac{\partial u_{y}}{\partial x} - \frac{\partial u_{x}}{\partial y}} = 0}} & (1597) \end{matrix}$

Thus, these certainly satisfy eqs. (18), (19). However, from the viewpoint of Gauss's divergence theorem (Gauss's divergence theorem)

∫_(S)div(v)=∫_(C) n·vdc,  (1598)

what is seen is different slightly. Whereas the left side of the equation becomes zero according to eq. (1596), the right side of the equation becomes Q. In order to make the left side of the equation Q, div(v)=Q·δ(x, y) may be satisfied. Therefore, recognizing the differential equations as

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 669} \right\rbrack & \; \\ {{{div}(v)} = {{\frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y}} = {Q \cdot {\delta \left( {x,y} \right)}}}} & (1599) \\ {{{rot}(v)} = {{\frac{\partial u_{y}}{\partial x} - \frac{\partial u_{x}}{\partial y}} = 0}} & (1600) \end{matrix}$

is suitable for the representation of the source at the origin of coordinates. This format is included in eqs. (1548) and (1549). It should be noted that in eq. (1598), n represents an outward unit normal vector at a boundary C, and when a circular field is given, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 670} \right\rbrack & \; \\ {n = {\begin{Bmatrix} n_{x} \\ n_{y} \end{Bmatrix} = \begin{Bmatrix} {\cos \mspace{14mu} \theta} \\ {\sin \mspace{14mu} \theta} \end{Bmatrix}}} & (1601) \end{matrix}$

11.5.6 Solution of Vortex

Let the intensity (circulation) of vortex at the origin of coordinates be Γ[m²/s], and the velocity potential φ is given as follows according to eq. (1564):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 671} \right\rbrack & \; \\ {\varphi = {{\frac{\Gamma}{2\pi}\theta} = {\frac{\Gamma}{2\pi}\arctan \frac{y}{x}}}} & (1602) \end{matrix}$

Differentiating this gives a velocities of u_(x),u_(y), which are given as:

$\begin{matrix} {{{u_{x} \equiv \frac{\partial\varphi}{\partial x}} = {{- \frac{\Gamma}{2\pi}}\frac{y}{x^{2} + y^{2}}}}{{u_{y} \equiv \frac{\partial\varphi}{\partial y}} = {\frac{\Gamma}{2\pi}\frac{x}{x^{2} + y^{2}}}}} & (1603) \end{matrix}$

In the case where the intensity Γ=1, the vortex is called as a vortex of unit intensity. Let the velocity vector by the components u_(x),u_(y) of ea. (1603) be v, and we obtain:

$\begin{matrix} {{{div}(v)} = {{\frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y}} = 0}} & (1604) \\ {{{rot}(v)} = {{\frac{\partial u_{y}}{\partial x} - \frac{\partial u_{x}}{\partial y}} = 0}} & (1605) \end{matrix}$

Thus, these certainly satisfy eqs. (18), (19). However, from the viewpoint of Stokes' theorem (Stokes' theorem) in plane

[Formula 672]

∫_(S) rot(v)ds= _(C) t·vdc,  (1606)

what is seen is different slightly. Whereas the left side of the equation becomes zero according to eq. (1605), the right side of the equation becomes Γ. In order to make the left side of the equation Γ, rot(v)=Γ·δ(x,y) may be satisfied. Therefore, recognizing the differential equations as

$\begin{matrix} {{{div}\mspace{11mu} (v)} = {{\frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y}} = 0}} & (1607) \\ \left\lbrack {{Formula}\mspace{14mu} 673} \right\rbrack & \; \\ {{{rot}\mspace{11mu} (v)} = {{\frac{\partial u_{y}}{\partial x} - \frac{\partial u_{x}}{\partial y}} = {\Gamma \cdot {\delta \left( {x,y} \right)}}}} & (1608) \end{matrix}$

is suitable for the representation of the source at the origin of coordinates. This format is included in eqs. (1548) and (1549). It should be noted that in eq. (1606), t represents a unit tangent vector at a boundary C, and when a circular field is given, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 674} \right\rbrack & \; \\ {t = {\begin{Bmatrix} t_{x} \\ t_{y} \end{Bmatrix} = \begin{Bmatrix} {{- \sin}\mspace{11mu} \theta} \\ {\cos \mspace{11mu} \theta} \end{Bmatrix}}} & (1609) \end{matrix}$

11.5.7 States of Source and Vortex in Rectangular Region [Formula 675]

The observation region x and y given as

−a x a

−b y b,  (1610)

an aspect ratio κ is given as

$\begin{matrix} {\kappa \equiv {\frac{b}{a}.}} & (1611) \end{matrix}$

Let a characteristic length be R, and dimensionless coordinates {ξ,η} are given as:

$\begin{matrix} {\left\{ {\xi,\eta} \right\} \equiv \left\{ {\frac{x}{R},\frac{y}{R}} \right\}} & (1612) \end{matrix}$

Here, an observation region is given as:

$\begin{matrix} \left. {{- \frac{a}{R}}\mspace{14mu} \frac{x}{R}\mspace{14mu} \frac{a}{R}}\mspace{20mu}\Rightarrow\mspace{14mu} {{{- \frac{a}{R}}\mspace{14mu} \xi \mspace{14mu} \frac{a}{R}} - {\frac{b}{R}\mspace{14mu} \frac{y}{R}\mspace{14mu} \frac{b}{R}}}\mspace{14mu}\Rightarrow\mspace{14mu} {{- \kappa}\; \frac{a}{R}\mspace{14mu} \eta \mspace{14mu} \kappa \frac{a}{R}} \right. & (1613) \end{matrix}$

Defining a characteristic velocity owing to a source as Q/(2πR), and making the velocities u_(x), u_(y) according to eq. (1595) dimensionless, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 676} \right\rbrack & \; \\ {{{u_{xo} \equiv {u_{x}\text{/}\frac{Q}{2\pi \; R}}} = \frac{\xi}{\xi^{2} + \eta^{2}}}{{u_{yo} \equiv {u_{y}\text{/}\frac{Q}{2\pi \; R}}} = \frac{\eta}{\xi^{2} + \eta^{2}}}} & (1614) \end{matrix}$

The observation region is assumed to be in a square shape. A state of the dimensionless velocities is shown in FIG. 55.

Defining a characteristic velocity owing to a vortex as Γ(2πR), and making the velocities u_(x), u_(y) according to eq. (1603) dimensionless, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 677} \right\rbrack & \; \\ {{{u_{xo} \equiv {u_{x}\text{/}\frac{\Gamma}{2\pi \; R}}} = {- \frac{\eta}{\xi^{2} + \eta^{2}}}}{{u_{yo} \equiv {u_{y}\text{/}\frac{\Gamma}{2\pi \; R}}} = \frac{\xi}{\xi^{2} + \eta^{2}}}} & (1615) \end{matrix}$

The observation region is assumed to be in a square shape. A state of the dimensionless velocity is shown in FIG. 56.

11.5.8 Adjoint Differential Operator and Adjoint Boundary Condition [Formula 678]

The simultaneous differential equations (1548) and (1549) are generally described as eq. (23):,

$\begin{matrix} {{\sum\limits_{j}\; {L_{ij}u_{j}}} = f_{i}} & \underset{({Aforementioned})}{(23)} \end{matrix}$

Here, the original differential operators L_(ij) expressed by eq. (20):

$\begin{matrix} {\begin{bmatrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{bmatrix} \equiv \begin{bmatrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \\ {- \frac{\partial}{\partial y}} & \frac{\partial}{\partial x} \end{bmatrix}} & \underset{({Aforementioned})}{(20)} \end{matrix}$

The adjoint differential operators L_(ij)* are expressed by eq. (21):

$\begin{matrix} {\begin{bmatrix} L_{11}^{*} & L_{21}^{*} \\ L_{12}^{*} & L_{22}^{*} \end{bmatrix} \equiv \begin{bmatrix} {- \frac{\partial}{\partial x}} & \frac{\partial}{\partial y} \\ {- \frac{\partial}{\partial y}} & {- \frac{\partial}{\partial x}} \end{bmatrix}} & \underset{({Aforementioned})}{(21)} \end{matrix}$

A sum of integration obtained by multiplying the equation (23) by the dual velocity u_(i)*, that is, an inner product, is expressed by eq. (58):

$\begin{matrix} {{\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{L_{ij}{u_{j} \cdot u_{i}^{*}}\ {s}}}}} = {\sum\limits_{i}\; {\int_{S}{{f_{i} \cdot u_{i}^{*}}{s}}}}} & \underset{({Aforementioned})}{(58)} \end{matrix}$

Then, let the left side of the equation (58) be subjected to partial integration.

-   -   When [i=1,j=1], using eq. (816), we obtain:

$\begin{matrix} \begin{matrix} {{\int_{S}{L_{11}{u_{1} \cdot u_{1}^{*}}\ {s}}} \equiv {\int_{S}{{\frac{\partial u_{1}}{\partial x} \cdot u_{1}^{*}}\ {s}}}} \\ {= {{\int_{C}{n_{x}{u_{1} \cdot u_{1}^{*}}\ {c}}} - {\int_{S}{{u_{1} \cdot \frac{\partial u_{1}^{*}}{\partial x}}\ {s}}}}} \end{matrix} & (1616) \end{matrix}$

Let the boundary term of the right side of this equation be R₁₁, let the differential operator thereof be L₁₁*, and they are given as:

$\begin{matrix} {{R_{11} \equiv {\int_{C}{n_{x}{u_{1} \cdot u_{1}^{*}}\ {c}}}}{L_{11}^{*} \equiv {- \frac{\partial}{\partial x}}}} & (1617) \end{matrix}$

L₁₁* is the adjoint differential operator shown in eq. (21).

-   -   When [i=1,j=2], using eq. (817), we obtain:

$\begin{matrix} \begin{matrix} {{\int_{S}{L_{12}{u_{2} \cdot u_{1}^{*}}\ {s}}} \equiv {\int_{S}{{\frac{\partial u_{2}}{\partial y} \cdot u_{1}^{*}}\ {s}}}} \\ {= {{\int_{C}{n_{y}{u_{2} \cdot u_{1}^{*}}\ {c}}} - {\int_{S}{{u_{2} \cdot \frac{\partial u_{1}^{*}}{\partial y}}\ {s}}}}} \end{matrix} & (1618) \end{matrix}$

Let the boundary term of the right side of this equation be R₁₂, let the differential operator thereof be L₁₂*, and they are given as:

$\begin{matrix} {{R_{12} \equiv {\int_{C}{n_{y}{u_{2} \cdot u_{1}^{*}}{c}}}}{L_{12}^{*} \equiv {- \frac{\partial}{\partial y}}}} & (1619) \end{matrix}$

L₁₂* is the adjoint differential operator shown in eq. (21).

-   -   When [i=2,j=1], using eq. (817), we obtain:

$\begin{matrix} \begin{matrix} {{\int_{S}{L_{21}{u_{1} \cdot u_{2}^{*}}{s}}} \equiv {\int_{S}{{{- \frac{\partial u_{1}}{\partial y}} \cdot u_{2}^{*}}{s}}}} \\ {= {{- {\int_{C}{n_{y}{u_{1} \cdot u_{2}^{*}}{c}}}} + {\int_{S}{{u_{1} \cdot \frac{\partial u_{2}^{*}}{\partial y}}{s}}}}} \end{matrix} & (1620) \end{matrix}$

[Formula 679]

Let the boundary term of the right side of this equation be R₂₁, let the differential operator thereof be L₂₁*, and they are given as:

$\begin{matrix} {{R_{21} \equiv {- {\int_{C}{n_{y}{u_{1} \cdot u_{2}^{*}}{c}}}}}{L_{21}^{*} \equiv \frac{\partial}{\partial y}}} & (1621) \end{matrix}$

L₂₁* is the adjoint differential operator shown in eq. (21).

-   -   When [i=2,j=2], using eq. (816), we obtain:

$\begin{matrix} \begin{matrix} {{\int_{S}{L_{22}{u_{2} \cdot u_{2}^{*}}{s}}} \equiv {\int_{S}{{\frac{\partial u_{2}}{\partial x} \cdot u_{2}^{*}}{s}}}} \\ {= {{\int_{C}{n_{x}{u_{2} \cdot u_{2}^{*}}{c}}} - {\int_{S}{{u_{2} \cdot \frac{\partial u_{2}^{*}}{\partial x}}{s}}}}} \end{matrix} & (1622) \end{matrix}$

Let the boundary term of the right side of this equation be R₂₂, let the differential operator thereof be L₂₂*, and they are given as:

$\begin{matrix} {{R_{22} \equiv {\int_{C}{n_{x}{u_{2} \cdot u_{2}^{*}}{c}}}}{L_{22}^{*} \equiv {- \frac{\partial}{\partial x}}}} & (1623) \end{matrix}$

L₂₂* is the adjoint differential operator shown in eq. (21).

In the case where the following is satisfied as in the present case, the operators are referred to as non-self-adjoint differential operators:

L _(ji) *≠L _(ij)  (22) (Aforementioned)

Adding the boundary terms and giving it as R, we obtain:

$\begin{matrix} \begin{matrix} {R \equiv {R_{11} + R_{12} + R_{21} + R_{22}}} \\ {= {{\int_{C}{n_{x}{u_{1} \cdot u_{1}^{*}}{c}}} + {\int_{C}{n_{y}{u_{2} \cdot u_{1}^{*}}{c}}} - {\int_{C}{n_{y}{u_{1} \cdot u_{2}^{*}}{c}}} + {\int_{C}{n_{x}{u_{2} \cdot u_{2}^{*}}{c}}}}} \end{matrix} & (1624) \end{matrix}$

This equation is transformed to:

R=∫ _(C)(n _(x) u ₁ +n _(y) u ₂)u ₁ *dc+∫ _(C)(n _(x) u ₂ −n _(y) u ₁)u ₂ *dc  (1625)

Here, the following relationship between components n_(x),n_(y) of an outward unit normal vector and components t_(x),t_(y) of a unit tangent vector is used.

nx=t _(y)

n _(y) =−t _(x)  (1626)

[Formula 680]

Then, we obtain:

R=∫ _(C)(n _(x) u ₁ +n _(y) u ₂)u ₁ *dc+∫ _(C)(t _(x) u ₂ −t _(y) u ₂)u ₂ *dc  (1627)

Further, a normal direction velocity u_(n), and a tangent direction velocity u_(t) are given as:

u _(n) =n _(x) u ₁ +n _(y) u ₂

u _(t) =t _(x) u ₁ +t _(y) u ₂  (1628)

Using this, we obtain:

R=∫ _(C)(u _(n) ·u ₁ *+u _(t) ·u ₂*)dc  (1629)

According to the above-described results, partial integration of the left side of eq. (58) gives:

$\begin{matrix} {{\sum\limits_{i}{\sum\limits_{j}{\int_{S}{L_{ij}{u_{j} \cdot u_{j}^{*}}{s}}}}} = {R + {\sum\limits_{i}{\sum\limits_{j}{\int_{S}{{u_{j} \cdot L_{uj}^{*}}u_{i}^{*}{s}}}}}}} & \underset{({Aforementioned})}{(59)} \end{matrix}$

Here, details of the integrand of the right side of the equation are described below:

$\begin{matrix} {{{\begin{Bmatrix} u_{1} & u_{2} \end{Bmatrix}\begin{bmatrix} L_{11}^{*} & L_{21}^{*} \\ L_{12}^{*} & L_{22}^{*} \end{bmatrix}}\begin{Bmatrix} u_{1}^{*} \\ u_{2}^{*} \end{Bmatrix}} \equiv {{\begin{Bmatrix} u_{1} & u_{2} \end{Bmatrix}\begin{bmatrix} {- \frac{\partial}{\partial x}} & \frac{\partial}{\partial y} \\ {- \frac{\partial}{\partial y}} & {- \frac{\partial}{\partial x}} \end{bmatrix}}\begin{Bmatrix} u_{1}^{*} \\ u_{2}^{*} \end{Bmatrix}}} & (1630) \end{matrix}$

Extracting work of the adjoint differential operators L_(ij)*, we obtain:

$\begin{matrix} {{\begin{bmatrix} L_{11}^{*} & L_{21}^{*} \\ L_{12}^{*} & L_{22}^{*} \end{bmatrix}\begin{Bmatrix} u_{1}^{*} \\ u_{2}^{*} \end{Bmatrix}} \equiv \begin{Bmatrix} {\frac{\partial u_{2}^{*}}{\partial y} - \frac{\partial u_{1}^{*}}{\partial x}} \\ {{- \frac{\partial u_{2}^{*}}{\partial x}} - \frac{\partial u_{1}^{*}}{\partial y}} \end{Bmatrix}} & (1631) \end{matrix}$

Further, the dual problem of eq. (65) is given as:

$\begin{matrix} {{\sum\limits_{j}{L_{ji}^{*}u_{j}^{*}}} = f_{i}^{*}} & \underset{({Aforementioned})}{(65)} \end{matrix}$

f_(i)* of the right side of the equation are external force terms of the dual problem. Therefore, according to eq. (1631), the dual problem (65) is transformed to:

$\begin{matrix} {{\frac{\partial u_{2}^{*}}{\partial y} - \frac{\partial u_{1}^{*}}{\partial x}} = f_{1}^{*}} & (1632) \\ \left\lbrack {{Formula}\mspace{14mu} 681} \right\rbrack & \; \\ {{{- \frac{\partial u_{2}^{*}}{\partial x}} - \frac{\partial u_{1}^{*}}{\partial y}} = f_{2}^{*}} & (1633) \end{matrix}$

In eqs. (1632) and (1633), −u₂* is regarded as a velocity v_(x) in the x direction, and −u₁* is regarded as a velocity v_(y) in the y direction. They are replaced as follows:

(−u ₂*)→v _(x),

(−u ₁*)→v _(y),  (1634)

Then, we find that eq. (1632) expresses the vortex condition of eq. (1549), and eq. (1633) expresses the continuum condition of eq. (1548). As physical interpretation of the dual problem, here, −u₂* is regarded as a velocity in the x direction, and −u₁* is regarded as a velocity in the y direction. From this viewpoint, eq. (1624) can be transformed to:

R=∫ _(C) u ₁·(n _(x) u ₁ *−n _(y) u ₂*)dc+∫ _(C) u ₂·(n _(y) u ₁ *+n _(x) u ₂*)dc  (1635)

Here, using eq. (1626) gives:

R=∫ _(C) u ₁·(t _(x) u ₂ *+t _(y) u ₁*)dc+∫ _(C) u ₂·(n _(x) u ₂ *+n _(y) u ₁*)dc  (1636)

Further, the normal direction velocity u_(n)*, and the tangent direction velocity u_(t)* are given as follows according to the understanding about eq. (1634):

u _(n) *=n _(x) v _(x) +n _(y) v _(y) ≡−n _(x) u ₂ *−n _(y) u ₁*

u _(t) *=t _(x) v _(x) +t _(y) v _(y) ≡−t _(x) u ₂ *−t _(y) u ₁*  (1637)

With use of eq. (1637), eq. (1636) is transformed to:

R=−∫ _(C)(u ₁ ·u _(t) *+u ₂ ·u _(n)*)dc  (1638)

From this, we can see that the boundary term of eq. (59) can be understood in two ways according to eq. (1629) and according to eq. (1638), and the more convenient one may be used. What physical quantity the argument functions u₁*, u₂* of the dual problem are associated with is a matter decided by a person using the present solution method, as is the case with eq. (1643). Whatever physical quantity they are associated with, it does not affect the solution of the primal problem.

11.5.9 Homogenization of Boundary Condition and Boundary Term

An index B is added to a term that satisfies an inhomogeneous boundary condition so as to let the term be U_(Bj), and an index H is added to a term that satisfies a homogeneous boundary condition so as to let the term be u_(Hj). A primal velocity is expressed by a sum of these, which is given as:

[Formula 682]

u _(j) ≡u _(Bj) +u _(Hj)  (24) (Aforementioned)

Substituting this equation into eq. (23), we obtain the following simultaneous partial differential equation (40) with a homogeneous boundary condition:

$\begin{matrix} {{\sum\limits_{j}{L_{ij}u_{j}}} = f_{i}} & \underset{({Aforementioned})}{(23)} \\ {{\sum\limits_{j}{L_{ij}u_{Hj}}} = f_{Hi}} & \underset{({Aforementioned})}{(40)} \end{matrix}$

An inner product of this with the function u_(H)* satisfying the homogeneous adjoint boundary condition is given as:

$\begin{matrix} {{\sum\limits_{i}{\sum\limits_{j}{\int_{S}{L_{ij}{u_{Hj} \cdot u_{Hi}^{*}}{s}}}}} = {\sum\limits_{i}{\int_{S}{{f_{Hi} \cdot u_{Hi}^{*}}{s}}}}} & \underset{({Aforementioned})}{(61)} \end{matrix}$

Then, partial integration of the left side of the equation (61) gives:

-   -   When [i=1,j=1], using eq. (816), we obtain:

$\begin{matrix} \begin{matrix} {{\int_{S}{L_{11}{u_{H\; 1} \cdot u_{H\; 1}^{*}}{s}}} \equiv {\int_{S}{{\frac{\partial u_{H\; 1}}{\partial x} \cdot u_{H\; 1}^{*}}{s}}}} \\ {= {{\int_{C}{n_{x}{u_{H\; 1} \cdot u_{H\; 1}^{*}}{c}}} - {\int_{S}{{u_{H\; 1} \cdot \frac{\partial u_{H\; 1}^{*}}{\partial x}}{s}}}}} \end{matrix} & (1639) \end{matrix}$

Let the boundary term of the right side of this equation be R₁₁, let the differential operator thereof be L₁₁*, and they are given as:

$\begin{matrix} {{R_{H\; 11} \equiv {\int_{C}{n_{x}{u_{H\; 1} \cdot u_{H\; 1}^{*}}{c}}}}{L_{11}^{*} \equiv {- \frac{\partial}{\partial x}}}} & (1640) \end{matrix}$

L₁₁* is the adjoint differential operator shown in eq. (21).

-   -   When [i=1,f=2], using eq. (817), we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 683} \right\rbrack & \; \\ \begin{matrix} {{\int_{S}{L_{12}{u_{H\; 2} \cdot u_{H\; 1}^{*}}\ {s}}} \equiv {\int_{S}{{\frac{\partial u_{H\; 2}}{\partial y} \cdot u_{H\; 1}^{*}}{s}}}} \\ {= {{\int_{C}{n_{y}{u_{H\; 2} \cdot u_{H\; 1}^{*}}\ {c}}} - {\int_{S}{{u_{H\; 2} \cdot \frac{\partial u_{H\; 1}^{*}}{\partial y}}\ {s}}}}} \end{matrix} & (1641) \end{matrix}$

Let the boundary term of the right side of this equation be R₁₂ let the differential operator thereof be L₁₂*, and they are given as:

$\begin{matrix} {{R_{H\; 12} \equiv {\int_{C}{n_{y}{u_{H\; 2} \cdot u_{H\; 1}^{*}}\ {c}}}}{L_{12}^{*} \equiv {- \frac{\partial}{\partial y}}}} & (1642) \end{matrix}$

L₁₂* is the adjoint differential operator given in eq. (21).

-   -   When [i=2,j=1], using eq. (817), we obtain:

$\begin{matrix} \begin{matrix} {{\int_{S}{L_{21}{u_{H\; 1} \cdot u_{H\; 2}^{*}}\ {s}}} \equiv {\int_{S}{{{- \frac{\partial u_{H\; 1}}{\partial y}} \cdot u_{H\; 2}^{*}}\ {s}}}} \\ {= {{- {\int_{C}{n_{y}{u_{H\; 1} \cdot u_{H\; 2}^{*}}\ {c}}}} + {\int_{S}{{u_{H\; 1} \cdot \frac{\partial u_{H\; 2}^{*}}{\partial y}}\ {s}}}}} \end{matrix} & (1643) \end{matrix}$

Let the boundary term of the right side of this equation be R₂₁, let the differential operator thereof be L₂₁*, and they are given as:

$\begin{matrix} {{R_{H\; 21} \equiv {- {\int_{C}{n_{y}{u_{1} \cdot u_{2}^{*}}\ {c}}}}}{L_{21}^{*} \equiv \frac{\partial}{\partial y}}} & (1644) \end{matrix}$

L₂₁* is the adjoint differential operator given in eq. (21).

-   -   When [i=2,j=2], using eq. (816), we obtain:

$\begin{matrix} \begin{matrix} {{\int_{S}{L_{22}{u_{2} \cdot u_{2}^{*}}\ {s}}} \equiv {\int_{S}{{\frac{\partial u_{2}}{\partial x} \cdot u_{2}^{*}}\ {s}}}} \\ {= {{\int_{C}{n_{x}{u_{2} \cdot u_{2}^{*}}\ {c}}} - {\int_{S}{{u_{2} \cdot \frac{\partial u_{2}^{*}}{\partial x}}\ {s}}}}} \end{matrix} & (1645) \end{matrix}$

Let the boundary term of the right side of this equation be R₂₂, let the differential operator thereof be L₂₂*, and they are given as:

$\begin{matrix} {{R_{H\; 22} \equiv {\int_{C}{n_{x}{u_{2} \cdot u_{2}^{*}}\ {c}}}}{L_{22}^{*} \equiv {- \frac{\partial}{\partial x}}}} & (1646) \end{matrix}$

L₂₂* is an adjoint differential operator given in eq. (21).

[Formula 684]

In the case where the following is satisfied as in the present case, the operators are referred to as non-self-adjoint differential operators.

L* _(ji) ≠L _(ij)  (22) (Aforementioned)

Adding the boundary terms and expressing the same as R_(H), we obtain:

$\begin{matrix} \begin{matrix} {R_{H} \equiv {R_{H\; 11} + R_{H\; 12} + R_{H\; 21} + R_{H\; 22}}} \\ {= {{\int_{C}{n_{x}{u_{H\; 1} \cdot u_{H\; 1}^{*}}\ {c}}} + {\int_{C}{n_{y}{u_{H\; 2} \cdot u_{H\; 1}^{*}}\ {c}}}}} \\ {{- {\int_{C}{n_{y}{u_{H\; 1} \cdot u_{H\; 2}^{*}}\ {c}}}} + {\int_{C}{n_{x}{u_{H\; 2} \cdot u_{H\; 2}^{*}}\ {c}}}} \end{matrix} & (1647) \end{matrix}$

Transforming this equation gives:

R _(H)=∫_(C)(n _(x) u _(H1) +n _(y) u _(H2))·u _(H1) *dc+∫ _(C)(n _(x) u _(H2) −n _(y) u _(H1))·u _(H2) *dc  (1648)

Here, using eq. (1626), we obtain:

R _(H)=∫_(C)(n _(x) u _(H1) +n _(y) u _(H2))·u _(H1) *dc+∫ _(C)(t _(x) u _(H1) −t _(y) u _(H2))·u _(H2) *dc  (1649)

Further, using the normal direction velocity u_(Hn), and the tangent direction velocity u_(Ht) satisfying the following

u _(Hn) =n _(x) u _(H1) +n _(y) u _(H2)

u _(Ht) =t _(x) u _(H1) +t _(y) u _(H2),  (1650)

we obtain:

R _(H)=∫_(C)(u _(Hn) ·u _(H1) *+u _(Ht) ·u _(H2)*)dc  (1651)

According to the results of the above-described operation, partial integration of the left side of eq. (61) gives:

$\begin{matrix} {{\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{L_{ij}{u_{Hj} \cdot u_{Hi}^{*}}\ {s}}}}} = {R_{H} + {\sum\limits_{i}\; {\sum\limits_{j}\; {\int_{S}{{u_{Hj} \cdot L_{ij}^{*}}u_{Hi}^{*}\ {s}}}}}}} & \underset{({Aforementioned})}{(62)} \end{matrix}$

Here, details of the integrand of the right side of the equation are described below:

$\begin{matrix} {{{\left\{ {u_{H\; 1}\mspace{14mu} u_{H\; 2}} \right\} \begin{bmatrix} L_{11}^{*} & L_{21}^{*} \\ L_{12}^{*} & L_{22}^{*} \end{bmatrix}}\begin{Bmatrix} u_{H\; 1}^{*} \\ u_{H\; 2}^{*} \end{Bmatrix}} \equiv {{\left\{ {u_{H\; 1}\mspace{14mu} u_{H\; 2}} \right\} \begin{bmatrix} {- \frac{\partial}{\partial x}} & \frac{\partial}{\partial y} \\ {- \frac{\partial}{\partial y}} & {- \frac{\partial}{\partial x}} \end{bmatrix}}\begin{Bmatrix} u_{H\; 1}^{*} \\ u_{H\; 2}^{*} \end{Bmatrix}}} & (1652) \end{matrix}$

Extracting work of the adjoint differential operators L_(ij)*, we obtain:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 685} \right\rbrack & \; \\ {{\begin{bmatrix} L_{11}^{*} & L_{21}^{*} \\ L_{12}^{*} & L_{22}^{*} \end{bmatrix}\begin{Bmatrix} u_{H\; 1}^{*} \\ u_{H\; 2}^{*} \end{Bmatrix}} \equiv \begin{Bmatrix} \frac{\partial u_{H\; 2}^{*}}{\partial y} & {- \frac{\partial u_{H\; 1}^{*}}{\partial x}} \\ {- \frac{\partial u_{H\; 2}^{*}}{\partial x}} & {- \frac{\partial u_{H\; 1}^{*}}{\partial y}} \end{Bmatrix}} & (1653) \end{matrix}$

The dual problem equation (66) with a homogeneous boundary condition is given as:

$\begin{matrix} {{\sum\limits_{j}\; {L_{ji}^{*}u_{Hj}^{*}}} = f_{Hi}^{*}} & \underset{({Aforementioned})}{(66)} \end{matrix}$

Here, f_(Hi)* of the right side of the equation represent the external force terms of the dual problem.

Therefore, according to eq. (1653), the dual problem equation (66) is transformed to:

$\begin{matrix} {{\frac{\partial u_{H\; 2}^{*}}{\partial y} - \frac{\partial u_{H\; 1}^{*}}{\partial x}} = f_{H\; 1}^{*}} & (1654) \\ {{{- \frac{\partial u_{H\; 2}^{*}}{\partial x}} - \frac{\partial u_{H\; 1}^{*}}{\partial y}} = f_{H\; 2}^{*}} & (1655) \end{matrix}$

In eqs. (1654) and (1655), −u_(H2)* is regarded as a velocity v_(x) in the x direction, and −u_(H1)* is regarded as a velocity v_(y) in the y direction. They are replaced as follows:

(−u _(H2)*)→v _(x)

(−u _(H1)*)→v _(y),  (1656)

Then, we find that eq. (1654) expresses the vortex condition of eq. (1549), and eq. (1655) expresses the continuum condition of eq. (1548). As physical interpretation of the dual problem, here, −u_(H2)* is regarded as a velocity in the x direction, and −u_(H1)*is regarded as a velocity in the y direction. From this viewpoint, eq. (1647) is transformed to:

[Formula(686]

R _(H)=∫_(C) u _(H1)·(n _(x) u _(H1) −n _(y) u _(H2)*)dc+∫ _(C) u _(H2)·(n _(y) u _(H1) *+n _(x) u _(H2)*)dc  (1657)

Here, using eq. (1626) gives:

R _(H)=∫_(C) u _(H1)·(t _(x) u _(H2) −t _(y) u _(H1)*)dc+∫ _(C) u _(H2)·(n _(x) u _(H2) *+n _(y) u _(H1)*)dc  (1658)

Further, the normal direction velocity u_(Hn)*, and the tangent direction velocity u_(Ht)* are given as follows according to the understanding about eq. (1656):

u _(Hn) *=n _(x) v _(x) +n _(y) v _(y) ≡−n _(x) u _(H2) *−n _(y) u _(H1)*

u _(Ht) *=t _(x) v _(x) +t _(y) v _(y) ≡−t _(x) u _(H2) *−t _(y) u _(H1)*  (1659)

With use of eq. (1659), eq. (1658) is transformed to:

R _(H)=−∫_(C)(u _(H1) ·u _(Ht) *+u _(H2) ·u _(Hn)*)dc  (1660)

From this, we can see that the boundary term R_(H) of eq. (62) can be understood in two ways according to eq. (1651) and according to eq. (1660), and the more convenient one may be used. What physical quantity the argument functions u_(H1)*, u_(H2)* of the dual problem are associated with is a matter decided by a person using the present solution method, as is the case with eq. (1656). Whatever physical quantity they are associated with, it does not affect the solution of the primal problem.

11.5.10 Getting Equations on Each Function from the Simultaneous Eigenvalue Problem

R is assumed to be the characteristic length, and the weight constant is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 687} \right\rbrack & \; \\ {w_{1} = {w_{2} = {w \equiv \frac{1}{R}}}} & (1661) \end{matrix}$

Then, details of the following primal simultaneous differential equation (105) are given as eqs. (1662) and (1663):

$\begin{matrix} {{\sum\limits_{j}\; {L_{ij}\varphi_{j}}} = {\lambda \; w_{i}\varphi_{i}^{*}}} & \underset{({Aforementioned})}{(105)} \\ {{\frac{\partial\varphi_{x}}{\partial x} + \frac{\partial\varphi_{y}}{\partial y}} = {\lambda \; w\; \varphi_{x}^{*}}} & (1662) \\ {{{- \frac{\partial\varphi_{x}}{\partial y}} + \frac{\partial\varphi_{y}}{\partial x}} = {\lambda \; w\; \varphi_{y}^{*}}} & (1663) \end{matrix}$

On the other hand, details of the following dual simultaneous differential equation (106) are given as eqs. (1664) and (1665):

$\begin{matrix} {{\sum\limits_{j}\; {L_{ji}^{*}\varphi_{j}^{*}}} = {\lambda \; w_{i}\varphi_{i}}} & \underset{({Aforementioned})}{(106)} \\ {{{- \frac{\partial\varphi_{x}^{*}}{\partial x}} + \frac{\partial\varphi_{y}^{*}}{\partial y}} = {\lambda \; w\; \varphi_{x}}} & (1664) \\ \left\lbrack {{Formula}\mspace{14mu} 688} \right\rbrack & \; \\ {{{- \frac{\partial\varphi_{x}^{*}}{\partial y}} - \frac{\partial\varphi_{y}^{*}}{\partial x}} = {\lambda \; w\; \varphi_{y}}} & (1665) \end{matrix}$

The primal simultaneous eigenvalue problem is as expressed by eq. (98):

$\begin{matrix} {{\sum\limits_{j}\; {\sum\limits_{k}\; {L_{ji}^{*}\frac{1}{w_{j}}L_{jk}\varphi_{k}}}} = {\lambda^{2}w_{i}\varphi_{i}}} & \underset{({Aforementioned})}{(98)} \end{matrix}$

Transposing the weight constant w according to eq. (1661), we obtain:

$\begin{matrix} {{\sum\limits_{j}\; {\sum\limits_{k}\; {L_{ji}^{*}L_{jk}\varphi_{k}}}} = {\lambda^{2}w^{2}\varphi_{i}}} & (1666) \end{matrix}$

Using differential operators of eqs. (18), (19) we can calculate the left side of this equation, we obtain:

∇²φ_(x)=−λ² w ²φ_(x)  (1667)

∇²φ_(y)=−λ² w ²φ_(y)  (1668)

Successfully obtaining differential equations on each function from the simultaneous equations of eq. (1666), we obtain differential equations φ_(x),φ_(y), on which are functions φ satisfying the following:

(∇²+λ² w ²)φ=0  (1669)

On the other hand, the dual simultaneous eigenvalue problem is expressed by the following eq. (99):

$\begin{matrix} {{\sum\limits_{j}\; {\sum\limits_{k}\; {L_{ij}\frac{1}{w_{j}}L_{kj}^{*}\varphi_{k}^{*}}}} = {\lambda^{2}w_{i}\varphi_{i}^{*}}} & \underset{({Aforementioned})}{(99)} \end{matrix}$

Similarly this is transformed to

$\begin{matrix} {{\sum\limits_{j}\; {\sum\limits_{k}\; {L_{ij}L_{kj}^{*}\varphi_{k}^{*}}}} = {\lambda^{2}w^{2}\varphi_{i}^{*}}} & (1670) \end{matrix}$

Calculating the left side of this equation by using the differential operators of eqs. (18), (19), we obtain equations composed of the same operators as those in eqs. (1667), (1668) as follows:

∇²φ_(x)*=−λ² w ²φ_(x)*  (1671)

∇²φ_(y)*=−λ² w ²φ_(y)*  (1672)

Successfully obtaining differential equations on each function from the simultaneous equations of eq. (1670), we obtain differential equations on φ_(x)*, φ_(y)*, which are functions φ* satisfying the following:

(∇²+λ² w ²)φ*=0  (1673)

11.5.11 Solution Function

Since eqs. (1669) and (1673) are the same differential equation, consequently, we find that φ_(x), φ_(y), and φ_(x)*, φ_(y)* are composed of the same function set respectively. Therefore, focusing on solving eq. (1669), we give:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 689} \right\rbrack & \; \\ {{\omega \equiv {\lambda \; w}} = {\frac{1}{R}\lambda}} & (1674) \end{matrix}$

Then, the differential equation (1669) is transformed to:

$\begin{matrix} {{{\left( {\nabla^{2}{+ \omega^{2}}} \right)\varphi} \equiv {\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \omega^{2}} \right)\varphi}} = 0} & \underset{({Aforementioned})}{(1583)} \end{matrix}$

The solution by separation of variables of this equation is:

$\begin{matrix} {{{\varphi \left( {x,y} \right)} \equiv {{X(x)} \cdot {Y(y)}}} = {\begin{pmatrix} \sin & {\alpha \; x} \\ \cos & {\alpha \; x} \end{pmatrix} \times \begin{pmatrix} \sin & {\beta \; y} \\ \cos & {\beta \; y} \end{pmatrix}}} & \underset{({Aforementioned})}{(1593)} \end{matrix}$

Therefore, we obtain:

α²+β²=ω²  (1588) (Aforementioned)

According to eq. (1674), the primal simultaneous differential equations (1662), (1663) are transformed to:

$\begin{matrix} {{\frac{\partial\varphi_{x}}{\partial x} + \frac{\partial\varphi_{y}}{\partial y}} = {\omega\varphi}_{x}^{*}} & (1675) \\ {{{- \frac{\partial\varphi_{x}}{\partial y}} + \frac{\partial\varphi_{y}}{\partial x}} = {\omega\varphi}_{y}^{*}} & (1676) \end{matrix}$

The dual simultaneous differential equations (1664), (1665) are transformed to:

$\begin{matrix} {{{- \frac{\partial\varphi_{x}^{*}}{\partial x}} + \frac{\partial\varphi_{y}^{*}}{\partial y}} = {\omega\varphi}_{x}} & (1677) \\ {{{- \frac{\partial\varphi_{x}^{*}}{\partial y}} - \frac{\partial\varphi_{y}^{*}}{\partial x}} = {\omega\varphi}_{y}} & (1678) \end{matrix}$

According to the combinations of eq. (1593), the following eight solutions satisfy these primal and dual simultaneous differential equations at the same time:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 690} \right\rbrack & \; \\ (1) & \; \\ {F_{{SS}\; 1} \equiv \left\{ \begin{matrix} {\varphi_{x} = {\frac{\alpha}{\omega}\sin \mspace{11mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \\ {\varphi_{y} = {\frac{\beta}{\omega}\cos \mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \\ {\varphi_{x}^{*} = {\cos \mspace{14mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \\ {\varphi_{y}^{*} = 0} \end{matrix} \right.} & (1679) \\ (2) & \; \\ {F_{{AA}\; 1} \equiv \left\{ \begin{matrix} {\varphi_{x} = {\frac{\alpha}{\omega}\cos \mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \\ {\varphi_{y} = {\frac{\beta}{\omega}\sin \mspace{11mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \\ {\varphi_{x}^{*} = {{- \sin}\mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \\ {\varphi_{y}^{*} = 0} \end{matrix} \right.} & (1680) \\ (3) & \; \\ {F_{{SA}\; 1} \equiv \left\{ \begin{matrix} {\varphi_{x} = {{- \frac{\alpha}{\omega}}\cos \mspace{11mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \\ {\varphi_{y} = {\frac{\beta}{\omega}\sin \mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \\ {\varphi_{x}^{*} = {\sin \mspace{11mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \\ {\varphi_{y}^{*} = 0} \end{matrix} \right.} & (1681) \\ (4) & \; \\ {F_{{AS}\; 1} \equiv \left\{ \begin{matrix} {\varphi_{x} = {\frac{\alpha}{\omega}\sin \mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \\ {\varphi_{y} = {{- \frac{\beta}{\omega}}\cos \mspace{11mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \\ {\varphi_{x}^{*} = {\cos \mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \\ {\varphi_{y}^{*} = 0} \end{matrix} \right.} & (1682) \\ \left\lbrack {{Formula}\mspace{14mu} 691} \right\rbrack & \; \\ (5) & \; \\ {F_{{SS}\; 2} \equiv \left\{ \begin{matrix} {\varphi_{x} = {\frac{\beta}{\omega}\sin \mspace{11mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \\ {\varphi_{y} = {{- \frac{\alpha}{\omega}}\cos \mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \\ {\varphi_{x}^{*} = 0} \\ {\varphi_{y}^{*} = {\sin \mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \end{matrix} \right.} & (1683) \\ (6) & \; \\ {F_{{AA}\; 2} \equiv \left\{ \begin{matrix} {\varphi_{x} = {{- \frac{\beta}{\omega}}\cos \mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \\ {\varphi_{y} = {\frac{\alpha}{\omega}\sin \mspace{11mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \\ {\varphi_{x}^{*} = 0} \\ {\varphi_{y}^{*} = {\cos \mspace{11mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \end{matrix} \right.} & (1684) \\ (7) & \; \\ {F_{{SA}\; 2} \equiv \left\{ \begin{matrix} {\varphi_{x} = {\frac{\beta}{\omega}\cos \mspace{11mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \\ {\varphi_{y} = {\frac{\alpha}{\omega}\sin \mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \\ {\varphi_{x}^{*} = 0} \\ {\varphi_{y}^{*} = {\cos \mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \end{matrix}\; \right.} & (1685) \\ (8) & \; \\ {F_{{AS}\; 2} \equiv \left\{ \begin{matrix} {\varphi_{x} = {\frac{\beta}{\omega}\sin \mspace{11mu} \alpha \; x\mspace{11mu} \sin \mspace{11mu} \beta \; y}} \\ {\varphi_{y} = {\frac{\alpha}{\omega}\cos \mspace{11mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \\ {\varphi_{x}^{*} = 0} \\ {\varphi_{y}^{*} = {{- \sin}\mspace{11mu} \alpha \; x\mspace{11mu} \cos \mspace{11mu} \beta \; y}} \end{matrix} \right.} & (1686) \end{matrix}$

A solution such that the eigenvalue w of the primal simultaneous differential equations (1675), (1676) is zero is a homogeneous solution. According to the following combinations of solutions of the harmonic equation

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 692} \right\rbrack & \; \\ {{{{\varphi \left( {x,y} \right)} \equiv {{X(x)} \cdot {Y(y)}}} = {\begin{pmatrix} \sin & {vx} \\ \cos & {vx} \end{pmatrix} \times \begin{pmatrix} \sinh & {vy} \\ \cosh & {vy} \end{pmatrix}}},} & \underset{({Aforementioned})}{(1577)} \end{matrix}$

the following four homogeneous solutions of the primal simultaneous differential equations are obtained.

Here, v is an arbitrary constant.

$\begin{matrix} (1) & \; \\ {H_{{SS}\; 1} \equiv \left\{ \begin{matrix} {\varphi_{x} = {{- \sin}\mspace{14mu} {vx}\mspace{14mu} \cosh \mspace{14mu} {vy}}} \\ {\varphi_{y} = {\cos \mspace{14mu} {vx}\mspace{14mu} \sinh \mspace{14mu} {vy}}} \end{matrix} \right.} & (1687) \\ (2) & \; \\ {H_{{AA}\; 1} \equiv \left\{ \begin{matrix} {\varphi_{x} = {\cos \mspace{11mu} {vx}\mspace{14mu} \sinh \mspace{14mu} {vy}}} \\ {\varphi_{y} = {\sin \mspace{14mu} {vx}\mspace{14mu} \cosh \mspace{14mu} {vy}}} \end{matrix} \right.} & (1688) \\ \left\lbrack {{Formula}\mspace{14mu} 693} \right\rbrack & \; \\ (3) & \; \\ {H_{{SA}\; 1} \equiv \left\{ \begin{matrix} {\varphi_{x} = {\cos \mspace{14mu} {vx}\mspace{14mu} \cosh \mspace{14mu} {vy}}} \\ {\varphi_{y} = {\sin \mspace{14mu} {vx}\mspace{14mu} \sinh \mspace{14mu} {vy}}} \end{matrix} \right.} & (1689) \\ (4) & \; \\ {H_{{AS}\; 1} \equiv \left\{ \begin{matrix} {\varphi_{x} = {{- \sin}\mspace{14mu} {vx}\mspace{14mu} \sinh \mspace{14mu} {vy}}} \\ {\varphi_{y} = {\cos \mspace{14mu} {vx}\mspace{14mu} \cosh \mspace{14mu} {vy}}} \end{matrix} \right.} & (1690) \end{matrix}$

According to the combinations of solutions given as eq. (1571) of the harmonic equation

$\begin{matrix} {{{{\varphi \left( {x,y} \right)} \equiv {{X(x)} \cdot {Y(y)}}} = {\begin{pmatrix} \sinh & {vx} \\ \cosh & {vx} \end{pmatrix} \times \begin{pmatrix} \sin & {vy} \\ \cos & {vy} \end{pmatrix}}},} & \underset{({Aforementioned})}{(1571)} \end{matrix}$

four homogeneous solutions of the primal simultaneous differential equations are obtained as follows, where v represents an arbitrary constant.

$\begin{matrix} (1) & \; \\ {H_{{SS}\; 2} \equiv \left\{ \begin{matrix} {\varphi_{x} = {{- \sinh}\mspace{14mu} {vx}\mspace{14mu} \cos \mspace{14mu} {vy}}} \\ {\varphi_{y} = {\cosh \mspace{14mu} {vx}\mspace{14mu} \sin \mspace{11mu} {vy}}} \end{matrix} \right.} & (1691) \\ (2) & \; \\ {H_{{AA}\; 2} \equiv \left\{ \begin{matrix} {\varphi_{x} = {\cosh \mspace{14mu} {vx}\mspace{14mu} \sin \mspace{14mu} {vy}}} \\ {\varphi_{y} = {\sinh \mspace{14mu} {vx}\mspace{14mu} \cos \mspace{14mu} {vy}}} \end{matrix} \right.} & (1692) \\ (3) & \; \\ {H_{{SA}\; 2} \equiv \left\{ \begin{matrix} {\varphi_{x} = {{- \cosh}\mspace{14mu} {vx}\mspace{14mu} \cos \mspace{14mu} {vy}}} \\ {\varphi_{y} = {\sinh \mspace{14mu} {vx}\mspace{14mu} \sin \mspace{14mu} {vy}}} \end{matrix} \right.} & (1693) \\ (4) & \; \\ {H_{{AS}\; 2} \equiv \left\{ \begin{matrix} {\varphi_{x} = {\sinh \mspace{14mu} {vx}\mspace{14mu} \sin \mspace{14mu} {vy}}} \\ {\varphi_{y} = {\cosh \mspace{14mu} {vx}\mspace{14mu} \cos \mspace{14mu} {vy}}} \end{matrix} \right.} & (1694) \end{matrix}$

Though a boundary condition has not been reflected yet on the foregoing solutions, the solutions satisfy the primal simultaneous differential equations (1675), (1676).

11.5.12 Homogenization of Boundary Condition

In Section 11.5.1, we find that regarding a problem of hydrodynamics as well, the simultaneous partial differential equations (1548), (1549) are transformed to eq. (23), and can be expressed as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 694} \right\rbrack & \; \\ {{\sum\limits_{j}\; {L_{ij}u_{j}}} = f_{i}} & \underset{({Aforementioned})}{(23)} \end{matrix}$

As is the case with Section 3.1, an index B is added to a term that satisfies an inhomogeneous boundary condition so as to let the term be u_(Bj), and an index H is added to a term that satisfies a homogeneous boundary condition so as to let the term be u_(Hj). A primal velocity u_(j) is expressed by a sum of these, which is given as:

u _(j) ≡u _(Bj) +u _(Hj)  (24) (Aforementioned)

This can be written regarding each component as follows:

u _(x) ≡u ₁ ≡u _(B1) +u _(H1)

u _(y) ≡u ₂ ≡u _(B2) +u _(H2)  (1695)

u_(Bj) does not necessarily satisfy eq. (23), and when taking the form of eq. (24), it satisfies eq. (23). Substituting eq. (24) into eq. (23), we obtain simultaneous partial differential equations expressed with a homogeneous boundary condition as follows:

$\begin{matrix} {{\sum\limits_{j}\; {L_{ij}u_{Hj}}} = f_{Hi}} & \underset{({Aforementioned})}{(40)} \end{matrix}$

The external force term f_(H) is given as:

$\begin{matrix} {f_{Hi} \equiv {f_{i} - {\sum\limits_{j}{L_{ij}u_{Bj}}}}} & \begin{matrix} {\mspace{146mu} (41)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

Here, defining the boundary function differential term f_(Bi) as

$\begin{matrix} {{f_{Bi} \equiv {\sum\limits_{j}{L_{ij}u_{Bj}}}},} & (1696) \end{matrix}$

we obtain:

f _(Hi) ≡f _(i) −f _(Bi)  (1697)

[Source]

When whether or not flow distribution caused by a source existing at the origin of coordinates can be reproduced by the eigenfunction method is to be confirmed, what is actually done is to solve eqs. (1599) and (1600) as the simultaneous partial differential equation (23):

[Formula  695] $\begin{matrix} {{{div}(v)} = {{\frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y}} = {Q \cdot {\delta \left( {x,y} \right)}}}} & \begin{matrix} {\mspace{124mu} (1599)} \\ ({Aforementioned}\;) \end{matrix} \\ {{{rot}(v)} = {{\frac{\partial u_{y}}{\partial x} - \frac{\partial u_{x}}{\partial y}} = 0}} & \begin{matrix} {\mspace{124mu} (1600)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

The external force term f_(i) of eq. (23) can be written regarding each component as follows:

f _(x) ≡f ₁ ≡Q·δ(x,y)

f _(y) ≡f ₂≡0  (1698)

The condition necessary as the boundary function u_(Bj) is a condition of having a velocity different from that of eq. (1595) in a rectangular region and the same velocity as that of eq. (1595) only on the boundary. In the case where eq. (1595) is used as the boundary function u_(Bj), which satisfies the differential equations (1599), (1600), the boundary function differential term f_(Bi) coincides with the external force term f_(i), and the external force term f_(H) becomes zero. As a result, u_(Hj) is zero, and this is not an object to which the eigenfunction method is to be applied.

Here, u_(Bj) is not given, but u_(Bj) is used as the homogeneous solution in which the right side of the differential equation (23) is zero. As a result, the boundary function differential term f_(Bi) is zero, and the external force term f_(H) coincides with f_(i). Therefore, u_(Hj) is a particular solution of the differential equation (23) having the right-hand side of eq. (1698). In other words, the velocity of eq. (1595) is separated into the particular solution part u_(HJ) and the homogeneous solution part u_(Bj).

[Vortex]

When whether or not flow distribution caused by a vortex existing at the origin of coordinates can be reproduced by the eigenfunction method is to be confirmed, what is actually done is to solve eqs. (1607) and (1608) as the simultaneous partial differential equation (23):

$\begin{matrix} {{{div}(v)} = {{\frac{\partial u_{x}}{\partial x} + \frac{\partial u_{y}}{\partial y}} = 0}} & \begin{matrix} {\mspace{124mu} (1607)} \\ ({Aforementioned}\;) \end{matrix} \\ {{{rot}(v)} = {{\frac{\partial u_{y}}{\partial x} - \frac{\partial u_{x}}{\partial y}} = {\Gamma \cdot {\delta \left( {x,y} \right)}}}} & \begin{matrix} {\mspace{124mu} (1608)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

The external force term f_(i) of eq. (23) can be written regarding each component as follows:

f _(x) ≡f ₁≡0

fy≡f ₂≡Γ·δ(x,y)  (1699)

[Formula 696]

The condition necessary as the boundary function u_(Bj) is a condition of having a velocity different from that of eq. (1603) in a rectangular region and the same velocity as that of eq. (1603) only on the boundary. In the case where eq. (1603) is used as the boundary function u_(Bj), which satisfies the differential equations (1607), (1608), the boundary function differential term f_(Bi) coincides with the external force term f_(i), and the external force term f_(H) becomes zero. As a result, u_(Hj) is zero, and this is not an object to which the eigenfunction method is to be applied.

Here, u_(Bj) is not given, but u_(Bj) is used as the homogeneous solution in which the right side of the differential equation (23) is zero. As a result, the boundary function differential term f_(Bi) is zero, and the external force term f_(H) coincides with f_(i). Therefore, u_(Hj) is a particular solution of the differential equation (23) having the right-hand side of eq. (1699). In other words, the velocity of eq. (1603) is separated into the particular solution part u_(Hj) and the homogeneous solution part u_(Bj).

Since a similar eigenfunction method to that described in Section 5.1.1 is used, the outline is described herein. The solution function u_(Hj) is expressed by a sum of eigenfunctions as follows:

$\begin{matrix} {u_{Hj} \equiv {\sum\limits_{k}{c_{k}\varphi_{jk}}}} & \begin{matrix} {\mspace{146mu} (69)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

Substituting eq. (69) into the differential equation (40) gives:

$\begin{matrix} {{\sum\limits_{k}{c_{k}\lambda_{k}w_{i}\varphi_{ik}^{*}}} = f_{Hi}} & \begin{matrix} {\mspace{135mu} (109)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

Determining an inner product with the function φ_(i)* and utilizing orthogonality, we obtain:

$\begin{matrix} {{c_{k}\lambda_{k}{\sum\limits_{i}{\int_{S}{w_{i}{\varphi_{ik}^{*} \cdot \varphi_{ik}^{*}}\ {s}}}}} = {\sum\limits_{i}{\int_{S}{{f_{Hi} \cdot \varphi_{ik}^{*}}\ {s}}}}} & \begin{matrix} {\mspace{135mu} (110)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

This settles the coefficient c_(k).

The above-described equation (109) is characterized in that the external force term f_(H) is expressed by the dual eigenfunction φ_(i)*. Here, the case where boundary function differential term f_(Bi) is zero

[Formula 697]

and the external force term f_(H) coincides with f_(i) is dealt with, the boundary condition of the dual eigenfunction φ_(i)* should coincide with the external force term f_(i). Therefore, according to eqs. (1698) and (1699), we can see that the boundary condition of φ_(i)*, as well as the x,y components are zero on the peripheries.

The above-described studies prove that the boundary condition imposed on the eigenfunctions is given as:

$\begin{matrix} (1) & \; \\ \begin{matrix} {{\varphi_{x}^{*} = 0},{\varphi_{y}^{*} = 0}} & {{\text{:}{on}\mspace{14mu} x} = {\pm a}} \\ {{\varphi_{x}^{*} = 0},{\varphi_{y}^{*} = 0}} & {{\text{:}{on}\mspace{14mu} y} = {\pm b}} \end{matrix} & (1700) \end{matrix}$

This condition makes the boundary term R_(H) of the following equation (1651) zero.

R _(H)=∫_(C)(u _(Hn) ·u _(H1) *+u _(Ht) ·u _(H2)*)dc  (1651) (Aforementioned)

Therefore, according to Section 4.3, the eigenfunctions are obtained as an orthogonal function system.

11.5.13 Rules for Eigenfunction Names

In the eigenfunction shown in the next section, the eigenvalues α, β are formed with the combinations given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 698} \right\rbrack & \; \\ {{\alpha = \left\{ {{\frac{{2m} - 1}{2\; a}\pi},{\frac{2m}{2a}\pi}} \right\}}{\beta = \left\{ {{\frac{{2n} - 1}{2b}\pi},{\frac{2n}{2b}\pi}} \right\}}} & (1701) \end{matrix}$

where m,n are integers. There are two kinds of the eigenvalue α, and two kinds of the eigenvalue β. For example, a function F formed with the eigenvalues of

$\begin{matrix} {{\alpha = \left. {\frac{{2m} - 1}{2\; a}\pi}\Leftarrow(1) \right.}{\beta = \left. {\frac{2n}{2b}\pi}\Leftarrow(2) \right.}} & (1702) \end{matrix}$

is represented by “F₁₂”, which means that α of the type (1) and β of the type (2) are used. Here, the first index is the number indicative of the type of α, and the second index is the number indicative of the type of β.

Further, this is followed by an alphabet indicating the mode form (Section 11.4.20) exhibited by the primal eigenfunction. For example, if the function F is formed with the eigenvalue α of the type (1) and the eigenvalue β of the type (2), and is in the mode SS, the function is represented by F_(12SS) The boundary conditions that the eigenfunction satisfies are distinguished with numbers provided at upper right, such as “F_(12SS) ¹”. The number at upper right is the number indicative of the boundary condition.

11.5.14 Primal Eigenfunction and Dual Eigenfunction [Formula 699]

The solution functions F_(SS1), F_(AA1), F_(SA1), F_(AS1) and F_(SS2), F_(AA2), F_(SA2), F_(AS2) mentioned in Section 11.5.11 may be combined to obtain a solution that satisfies the following condition equations (1700):

$\begin{matrix} (1) & \; \\ \begin{matrix} {{\varphi_{x}^{*} = 0},{\varphi_{y}^{*} = 0}} & {{\text{:}{on}\mspace{14mu} x} = {\pm a}} \\ {{\varphi_{x}^{*} = 0},{\varphi_{y}^{*} = 0}} & {{\text{:}{on}\mspace{14mu} y} = {\pm b}} \end{matrix} & \underset{({Aforementoned})}{(1700)} \end{matrix}$

This results in that only giving appropriate eigenvalues α,β to each solution function is required, which is simple. More specifically, eight eigenfunctions shown below are provided.

Here, the functions are named according to the rules described in Section 11.5.13, and the boundary condition number is (1). λ is an eigenvalue in the primal simultaneous differential equations (1662), (1663) and the dual simultaneous differential equations (1664), (1665).

$\begin{matrix} {{(1)\mspace{11mu} F_{11\; {AA}}^{1}} \equiv \left\{ \begin{matrix} {{\alpha \; = \; {\frac{{2\; m}\; - \; 1}{2\; a}\; \pi}},\; {\beta \; = \; {\frac{{2\; n}\; - \; 1}{2\; b}\; \pi}},\; {\lambda \; = \; {R\; \sqrt{\; {\alpha^{2}\; + \; \beta^{2}}}}}} \\ {\varphi_{x}\; = \; {{- \frac{\beta}{\sqrt{a^{2} + \beta^{2}}}}\; \cos \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}\; = \; {\frac{\alpha}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}\; \sin \; \alpha \; x\; \cos \; \beta \; y}} \\ {{\varphi_{x}^{*}\; = \; 0},\; {\varphi_{y}^{*} = {\cos \; \alpha \; x\; \cos \; \beta \; y}}} \end{matrix} \right.} & (1703) \\ {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 700} \right\rbrack} & \; \\ {{(2)\mspace{14mu} F_{11\; {SS}}^{1}} \equiv \left\{ \begin{matrix} {{\alpha \; = \; {\frac{{2\; m}\; - \; 1}{2\; a}\; \pi}},\; {\beta \; = \; {\frac{{2\; n}\; - \; 1}{2\; b}\; \pi}},\; {\lambda \; = \; {R\; \sqrt{\; {\alpha^{2}\; + \; \beta^{2}}}}}} \\ {\varphi_{x}\; = \; {\frac{\alpha}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}\; \sin \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}\; = {\frac{\beta}{\sqrt{a^{2} + \beta^{2}}}\; \cos \; \alpha \; x\; \sin \; \beta \; y}} \\ {{\varphi_{x}^{*}\; = \; {\cos \; \alpha \; x\; \cos \; \beta \; y}},{\varphi_{y}^{*} = 0}} \end{matrix} \right.} & (1704) \\ {{(3)\mspace{14mu} F_{21\; {AS}}^{1}} \equiv \left\{ \begin{matrix} {{\alpha \; = \; {\frac{{2\; m}\;}{2\; a}\; \pi}},\; {\beta \; = \; {\frac{{2\; n}\; - \; 1}{2\; b}\; \pi}},\; {\lambda \; = \; {R\; \sqrt{\; {\alpha^{2}\; + \; \beta^{2}}}}}} \\ {\varphi_{x}\; = \; {\frac{\beta}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}\; \sin \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}\; = \; {\frac{\alpha}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}\; \cos \; \alpha \; x\; \cos \; \beta \; y}} \\ {{\varphi_{x}^{*}\; = \; 0},\; {\varphi_{y}^{*} = {{- \sin}\; \alpha \; x\; \cos \; \beta \; y}}} \end{matrix} \right.} & (1705) \\ {{(4)\mspace{14mu} F_{21\; {SA}}^{1}} \equiv \left\{ \begin{matrix} {{\alpha \; = \; {\frac{{2\; m}\;}{2\; a}\; \pi}},\; {\beta \; = \; {\frac{{2\; n}\; - \; 1}{2\; b}\; \pi}},\; {\lambda \; = \; {R\; \sqrt{\; {\alpha^{2}\; + \; \beta^{2}}}}}} \\ {\varphi_{x}\; = {{- \; \frac{\alpha}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}}\; \cos \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}\; = \; {\frac{\beta}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}\; \sin \; \alpha \; x\; \sin \; \beta \; y}} \\ {{\varphi_{x}^{*}\; = \; {\sin \; \alpha \; x\; \cos \; \beta \; y}},{\varphi_{y}^{*} = 0}} \end{matrix} \right.} & (1706) \\ {{(5)\mspace{14mu} F_{12\; {AS}}^{1}} \equiv \left\{ \begin{matrix} {{\alpha \; = \; {\frac{{2\; m}\; - \; 1}{2\; a}\; \pi}},\; {\beta \; = \; {\frac{2\; n}{2\; b}\; \pi}},\; {\lambda \; = \; {R\; \sqrt{\; {\alpha^{2}\; + \; \beta^{2}}}}}} \\ {\varphi_{x}\; = {\frac{\alpha}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}\; \sin \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}\; = {{- \; \frac{\beta}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}}\; \cos \; \alpha \; x\; \cos \; \beta \; y}} \\ {{\varphi_{x}^{*}\; = \; {\cos \; \alpha \; x\; \sin \; \beta \; y}},{\varphi_{y}^{*} = 0}} \end{matrix} \right.} & (1707) \\ {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 701} \right\rbrack} & \; \\ {{(6)\mspace{14mu} F_{12\; {SA}}^{1}} \equiv \left\{ \begin{matrix} {{\alpha \; = \; {\frac{{2\; m}\; - \; 1}{2\; a}\; \pi}},\; {\beta \; = \; {\frac{2\; n}{2\; b}\; \pi}},\; {\lambda \; = \; {R\; \sqrt{\; {\alpha^{2}\; + \; \beta^{2}}}}}} \\ {\varphi_{x}\; = {\frac{\beta}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}\; \cos \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}\; = {\frac{\alpha}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}\; \sin \; \alpha \; x\; \sin \; \beta \; y}} \\ {{\varphi_{x}^{*}\; = \; 0},\; {\varphi_{y}^{*}\cos \; \alpha \; x\; \sin \; \beta \; y}} \end{matrix} \right.} & (1708) \\ {\mspace{79mu} {{(7)\mspace{14mu} F_{22\; {AA}}^{1}} \equiv \left\{ \begin{matrix} {{\alpha \; = \; {\frac{2\; m}{2\; a}\; \pi}},\; {\beta \; = \; {\frac{2\; n}{2\; b}\; \pi}},\; {\lambda \; = \; {R\; \sqrt{\; {\alpha^{2}\; + \; \beta^{2}}}}}} \\ {\varphi_{x}\; = {\frac{\alpha}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}\; \cos \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}\; = {\frac{\beta}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}\; \sin \; \alpha \; x\; \cos \; \beta \; y}} \\ {{\varphi_{x}^{*}\; = {{- \sin}\; \alpha \; x\; \sin \; \beta \; y}},{\varphi_{y}^{*} = 0}} \end{matrix} \right.}} & (1709) \\ {\mspace{79mu} {{(8)\mspace{14mu} F_{22\; {SS}}^{1}} \equiv \left\{ \begin{matrix} {{\alpha \; = \; {\frac{2\; m}{2\; a}\; \pi}},\; {\beta \; = \; {\frac{2\; n}{2\; b}\; \pi}},\; {\lambda \; = \; {R\; \sqrt{\; {\alpha^{2}\; + \; \beta^{2}}}}}} \\ {\varphi_{x}\; = {\frac{\beta}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}\; \sin \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}\; = {{- \frac{\alpha}{\sqrt{\alpha^{2}\; + \; \beta^{2}}}}\; \cos \; \alpha \; x\; \sin \; \beta \; y}} \\ {{\varphi_{x}^{*}\; = \; 0},\; {\varphi_{y}^{*}\sin \; \alpha \; x\; \sin \; \beta \; y}} \end{matrix} \right.}} & (1710) \end{matrix}$

These eigenfunctions have orthogonality. Further, all of the eigenfunctions have orthogonality with respect to homogeneous solutions H_(SS1), H_(AA1), H_(SA1), H_(AS1) of eqs. (1687) to (1690), and also with respect to H_(SS2), H_(AA2), H_(SA2), H_(AS2) of eqs. (1691) to (1694). This means that a space representing a particular solution part of the velocity and a space representing a homogeneous solution part are orthogonal with each other. States of No. 1 mode of the primal eigenfunction and the dual eigenfunction of (1) to (8) are shown in FIGS. 57 to 64.

11.5.15 Eigenfunction Method

A solution is determined by the eigenfunction method described in Section 5.1. The determined solution is made dimensionless in the same manner as that described in Section 11.5.7, and states of flow are shown.

[Source] Given a square observation region, calculation was performed with the mode number mxn of the eigenfunction being set to at most 30×30. As the external force term f_(i) is given by eq. (1698), the particular solution part uH, can be expressed by the eigenfunction Fliiss of eq. (1704) alone. The state of dimensionless velocity is shown in FIG. 65. This result indicates the particular solution of eqs. (1599), (1600).

FIG. 65 is similar to the state determined by the analytical solution (FIG. 55), expressing a state of fluid springing out from the origin. This, however, shows only the particular solution part, and it is necessary to check how different it is from the analytical solution. The state obtained by subtracting the eigenfunction solution from the analytical solution of eq. (1595) is shown in FIG. 66.

FIG. 66 shows that difference is significant in the vicinities of the boundary. This is called “residual flow”. The residual flow is represented by u_(Bj) of the homogeneous solution part. The particular solution part is represented by the eigenfunction Fliiss of eq. (1704) alone, and the eigenvalue is given as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 702} \right\rbrack & \; \\ {{\alpha = {\frac{{2m} - 1}{2a}\pi}},{\beta = {\frac{{2n} - 1}{2b}\pi}}} & (1711) \end{matrix}$

Using this, as well as the homogeneous solution of the mode SS, H_(SS1) of eq. (1687), and H_(SS2) of eq. (1691), we express u_(Bj) as follows:

$\begin{matrix} {u_{Bj} \equiv {{\sum\limits_{m}{c_{1\; m}{H_{{SS}\; 1}\left( v_{m} \right)}}} + {\sum\limits_{m}{c_{2\; n}{H_{{SS}\; 2}\left( v_{n} \right)}}}}} & (1712) \end{matrix}$

where c_(1m),c_(2n) represent coefficients and v_(m),v_(n) represent values of v in the mode m,n. More specifically, the value of v of the homogeneous solution H_(SS1) is given in accordance with α as follows:

$\begin{matrix} {v_{m} \equiv {\frac{{2m} - 1}{2\; a}\pi}} & (1713) \end{matrix}$

The value of v of the homogeneous solution H_(SS2) is given in accordance with β as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 703} \right\rbrack & \; \\ {v_{n} = {\frac{{2n} - 1}{2b}\pi}} & (1714) \end{matrix}$

Since H_(SS1), H_(SS2) are orthogonal to F_(11SS) ¹, the coefficients c_(1m),c_(2n) can be determined by the least-squares method in which u_(Bj) is replaced with the velocity of eq. (1595). Results of calculation performed with the mode number mxn of the homogeneous solution being set to at most 30×30 are shown in FIG. 52.

FIG. 52 Velocity Distribution of Residual Flow for Source.

FIG. 52 is similar to FIG. 66, from which we can see that the residual flow can be expressed. In other words, this shows that the homogeneous solution part is determined appropriately.

[Formula 704]

A sum of the particular solution part ti (FIG. 65) and the homogeneous solution part u_(Bj) (FIG. 52) is shown in FIG. 53. This is the final result obtained by reproducing the analytical solutions of the source by the eigenfunction method.

FIG. 53 Velocity Distribution of (u_(Bj)+u_(Hj)) for Source.

From FIG. 53, we can see that the flow distribution by the source is reproduced substantially completely. In order to know final error distribution, a state obtained by subtracting (u_(Bj)+u_(Hj)) (FIG. 53) from the analytical solutions (FIG. 55) is shown in FIG. 54.

[Formula 705]

FIG. 54 Error of Velocity Distribution for Source.

From FIG. 54, we can see that errors remain at the origin and along the coordinate axes. These errors decrease as the mode number m×n of the eigenfunction is increased.

[Vortex]

Given a square observation region, calculation was performed with the mode number mxn of the eigenfunction being set to at most 30×30. As the external force term f_(i) is given by eq. (1699), the particular solution part u_(Hj) can be expressed by the eigenfunction F_(11AA) ¹ of eq. (1703) alone. The state of dimensionless velocity is shown in FIG. 55. This result indicates the particular solutions of eqs. (1607), (1608).

[Formula 706]

FIG. 55 Velocity Distribution by Eigenfunction Method for Vortex.

FIG. 55 is similar to the state determined by the analytical solution (FIG. 56), expressing fluid circulating around the origin as the center. This, however, shows only the particular solution part, and it is necessary to check how different it is from the analytical solution. The state obtained by subtracting the eigenfunction solution from the analytical solutions of eq. (1603) is shown in FIG. 56.

[Formula 707]

FIG. 56 Difference of Velocity Distribution between Analytical Solution and Eigenfunction Method for Vortex.

From FIG. 56, we can see that the difference is great in the vicinities of the boundary. This is called “residual flow”. The residual flow is represented by u_(Bj) of the homogeneous solution part. The particular solution part is represented by the eigenfunction F_(11AA) ¹ of eq. (1703) alone, and the eigenvalue is given as:

$\begin{matrix} {{\alpha = {\frac{{2m} - 1}{2a}\pi}},{\beta = {\frac{{2n} - 1}{2b}\pi}}} & (1715) \end{matrix}$

Using this, as well as the homogeneous solution of the mode AA, H_(AA1) of eq. (1688), and H_(AA2) of eq. (1692), we express u_(Bj) as follows:

$\begin{matrix} {u_{Bj} \equiv {{\sum\limits_{m}{c_{1\; m}{H_{{AA}\; 1}\left( v_{m} \right)}}} + {\sum\limits_{n}{c_{2\; n}{H_{{AA}\; 2}\left( v_{n} \right)}}}}} & (1716) \end{matrix}$

where c_(1m),c_(2n) represent coefficients and v_(m),v_(n) represent value of v in the mode m,n. More specifically, the value of v of the homogeneous solution H_(AA1) is given in accordance with α as follows:

$\begin{matrix} {v_{m} \equiv {\frac{{2\; m} - 1}{2\; a}\pi}} & (1717) \end{matrix}$

The value of v of the homogeneous solution H_(AA2) is given in accordance with β as follows:

$\begin{matrix} {v_{n} = {\frac{{2\; n} - 1}{2\; b}\pi}} & (1718) \end{matrix}$

[Formula 708]

Since H_(AA1),H_(AA2) are orthogonal to F_(11AA) ¹, the coefficients c_(1m),c_(2n) can be determined by the least squares method in which ud_(Bj) is replaced with the velocity of eq. (1603). Results of calculation performed with the mode number mxn of the homogeneous solution being set to at most 30×30 are shown in FIG. 57.

FIG. 57 Velocity Distribution of Residual Flow for Vortex.

FIG. 57 is similar to FIG. 56, from which we can see that the residual flow can be expressed. In other words, this shows that the homogeneous solution part u_(Bj) is determined appropriately.

A sum of the particular solution part u_(Hj) (FIG. 55) and the homogeneous solution part u_(Bj) (FIG. 57) is shown in FIG. 58. This is the final result obtained by reproducing the analytical solution of the vortex by the eigenfunction method.

[Formula 709]

FIG. 58 Velocity Distribution of (u_(Bj)+u_(Hj)) for Vortex.

From FIG. 58, we can see that the flow distribution by vortex is reproduced substantially completely. In order to know final error distribution, a state obtained by subtracting (u_(Bj)+u_(Hj)) (FIG. 58) from the analytical solution (FIG. 56) is shown in FIG. 59.

FIG. 59 Error of Velocity Distribution for Vortex.

From FIG. 59, we can see that errors remain at the origin and along the coordinate axes. These errors decrease as the mode number m×n of the eigenfunction is increased.

[Formula 710] 11.6 Bottom Wall/Uniform Flow at Top (Uniform Jet Flow)

This section should better be included in Section 10.

11.6.1 Problem Setting and Boundary Condition

In a two-dimensional irrotational flow, the origin is placed at the center of a rectangular region (2a×2b), and the periphery of the rectangle is the boundary. In this problem, the purpose is to know how the jet flow should be realized at right and left side edges (x=±a) in such boundary condition as the bottom wall (y=−b) is a rigid body and as the upward stream is uniform at the top side end (y=b). The bottom edge is a wall face, and we would like to know what flow should be caused to occur at the right and left edges in order that uniform flow occurs at the top edge.

According to the equation of continuity (18) and the vortex-free condition (19), the external force term f_(i) of eq. (23) is zero. This can be written regarding each component as follows:

f _(x) ≡f ₁≡0

f _(y) ≡f ₂≡0  (1719)

Let the velocity of uniform flow be v, and the boundary condition is given as:

u _(x)=0, u _(y) =V :on y=+b

u _(y)=0, :on y=−b  (1720)

According to the boundary integration term of eq. (1648), the following condition is imposed on the eigenfunction:

0=∫_(C)(n _(x)φ₁ +n _(y)φ₂)·φ₁ *dc+∫ _(C)(n _(x)φ₂ −n _(y)φ₁)·φ₂ *dc(1721)

[Formula 711]

Transforming this equation gives:

0=∫_(C) {n _(x)(φ₁·φ₁*+φ₂·φ₂*)+n _(y)(φ₂·φ₁*−φ₁·φ₂*)}dc  (1722)

Let the top edge (y=b) be named a “boundary C_(T)”, let the bottom edge (y=b) be named a “boundary c_(B)”, let the left edge (x=−a) be named a “boundary C_(L)”, and let the right edge (x=a) be named a “boundary C_(R)”. Then, the outward unit normal vector is given as:

$\begin{matrix} \left\{ {{\begin{matrix} {n_{x} = 0} \\ {n_{y} = 1} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{T}},\left\{ {\begin{matrix} {n_{x} = 0} \\ {n_{y} = {- 1}} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{B}\left\{ {{\begin{matrix} {n_{x} = {- 1}} \\ {n_{y} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{L}},\left\{ {\begin{matrix} {n_{x} = 1} \\ {n_{y} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{R}} \right.} \right.} \right.} \right. & (1723) \end{matrix}$

Substituting this equation into eq. (1722) gives:

$\begin{matrix} {0 = {{\int_{C_{T}}{\left( {{\varphi_{2} \cdot \varphi_{1}^{*}} - {\varphi_{1} \cdot \varphi_{2}^{*}}} \right)\ {c}}} - {\int_{C_{B}}{\left( {{\varphi_{2} \cdot \varphi_{1}^{*}} - {\varphi_{1} \cdot \varphi_{2}^{*}}} \right)\ {c}}} - {\int_{C_{L}}{\left( {{\varphi_{1} \cdot \varphi_{1}^{*}} - {\varphi_{2} \cdot \varphi_{2}^{*}}} \right)\ {c}}} + {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \varphi_{1}^{*}} - {\varphi_{2} \cdot \varphi_{2}^{*}}} \right)\ {c}}}}} & (1724) \end{matrix}$

In order to cause the homogeneous boundary condition of eq. (1720) to be reflected on this equation, the sign of zero is described under the three primal eigenfunctions. Then, we obtain:

$\begin{matrix} {0 = {{\int_{C_{T}}{\left( {{\underset{0}{\varphi_{2}} \cdot \varphi_{1}^{*}} - {\underset{0}{\varphi_{1}} \cdot \varphi_{2}^{*}}} \right)\ {c}}} - {\int_{C_{B}}{\left( {{\underset{0}{\varphi_{2}} \cdot \varphi_{1}^{*}} - {\varphi_{1} \cdot \varphi_{2}^{*}}} \right)\ {c}}} - {\int_{C_{L}}{\left( {{\varphi_{1} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \varphi_{2}^{*}}} \right)\ {c}}} + {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \varphi_{2}^{*}}} \right)\ {c}}}}} & (1725) \end{matrix}$

As to the term in which the primal eigenfunction does not have the sign of zero, no boundary condition is designated, and the value on the boundary is unknown. An adjoint boundary condition is decided so that integration value of each of the boundaries C_(T), C_(B), C_(L), C_(R) is zero, and we obtain:

$\begin{matrix} {0 = {{\int_{C_{T}}{\left( {{\underset{0}{\varphi_{2}} \cdot \varphi_{1}^{*}} - {\underset{0}{\varphi_{1}} \cdot \varphi_{2}^{*}}} \right)\ {c}}} - {\int_{C_{B}}{\left( {{\underset{0}{\varphi_{2}} \cdot \varphi_{1}^{*}} - {\varphi_{1} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}} - {\int_{C_{L}}{\left( {{\varphi_{1} \cdot \underset{0}{\varphi_{1}^{*}}} + {\varphi_{2} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}} + {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \underset{0}{\varphi_{1}^{*}}} + {\varphi_{2} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}}}} & (1726) \end{matrix}$

Thus, we obtain the following boundary condition imposed on the eigenfunction:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 712} \right\rbrack & \; \\ {\left( {}^{*} \right)\mspace{14mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{T}},\left\{ {\begin{matrix} {\varphi_{2} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{B}\mspace{40mu} \left\{ {{\begin{matrix} {\varphi_{1}^{*} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{L}},\left\{ {\begin{matrix} {\varphi_{1}^{*} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{R}} \right.} \right.} \right.} \right.} & (1727) \end{matrix}$

If the solution functions F_(SS1), F_(AA1), F_(SA1), F_(AS1) and F_(SS2), F_(AA2), F_(SA2), F_(AS2) described in Section 11.5.11 are combined and a solution satisfying the condition of eq. (1727) is obtained, it is the eigenfunction. However, unfortunately, an eigenfunction cannot be obtained in this function system. However, by changing the conditions at the boundaries C_(L),C_(R) slightly, we can obtain an eigenfunction of the following boundary condition:

$\begin{matrix} {(2)\mspace{14mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{T}},\left\{ {\begin{matrix} {\varphi_{2} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{B}\mspace{40mu} \left\{ {{\begin{matrix} {\varphi_{1}^{*} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{L}},\left\{ {\begin{matrix} {\varphi_{1}^{*} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{R}} \right.} \right.} \right.} \right.} & (1728) \end{matrix}$

Similarly, we can obtain an eigenfunction of the following boundary condition:

$\begin{matrix} {(3)\mspace{14mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{T}},\left\{ {\begin{matrix} {\varphi_{2} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{B}\mspace{40mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{L}},\left\{ {\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{R}} \right.} \right.} \right.} \right.} & (1729) \end{matrix}$

Further, we can obtain an eiaenfunction of the following boundary condition:

$\begin{matrix} {(4)\mspace{14mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{T}},\left\{ {\begin{matrix} {\varphi_{2} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{B}\mspace{40mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{L}},\left\{ {\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{R}} \right.} \right.} \right.} \right.} & (1730) \end{matrix}$

Only regarding (*), which is essential, an eigenfunction cannot be obtained. This is very interesting and suggestive. When only the integration term forms in the boundaries C_(L),C_(R) are extracted from eq. (1725), it is ideal that they satisfy:

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 713} \right\rbrack} & \; \\ {{\left( {}^{*} \right)\mspace{14mu} 0} = {{- {\int_{C_{L}}{\left( {{\varphi_{1} \cdot \underset{0}{\varphi_{1}^{*}}} + {\varphi_{2} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}}} + {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \underset{0}{\varphi_{1}^{*}}} + {\varphi_{2} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}}}} & (1731) \end{matrix}$

However, an eigenfunction that satisfy this equation is not obtained, and eigenfunctions satisfying the following are obtained:

$\begin{matrix} {{(2)\mspace{14mu} 0} = {{- {\int_{C_{L}}{\left( {{\varphi_{1} \cdot \underset{0}{\varphi_{1}^{*}}} + {\underset{0}{\varphi_{2}} \cdot \varphi_{2}^{*}}} \right)\ {c}}}} + {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \underset{0}{\varphi_{1}^{*}}} + {\underset{0}{\varphi_{2}} \cdot \varphi_{2}^{*}}} \right)\ {c}}}}} & (1732) \\ {{(3)\mspace{14mu} 0} = {{- {\int_{C_{L}}{\left( {{\underset{0}{\varphi_{1}} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}}} + {\int_{C_{R}}{\left( {{\underset{0}{\varphi_{1}} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}}}} & (1733) \\ {{(4)\mspace{14mu} 0} = {{- {\int_{C_{L}}{\left( {{\underset{0}{\varphi_{1}} \cdot \varphi_{1}^{*}} + {\underset{0}{\varphi_{2}} \cdot \varphi_{2}^{*}}} \right)\ {c}}}} + {\int_{C_{R}}{\left( {{\underset{0}{\varphi_{1}} \cdot \varphi_{1}^{*}} + {\underset{0}{\varphi_{2}} \cdot \varphi_{2}^{*}}} \right)\ {c}}}}} & (1734) \end{matrix}$

It is considered that since

$\underset{0}{\varphi_{2}} \cdot \varphi_{2}^{*}$

are included in (2) and (4), and

$\underset{0}{\varphi_{1}} \cdot \varphi_{1}^{*}$

are included in (3) and (4), these are canceled successfully as the number of applied modes increases, and the effect of (*) is exhibited by the

${\varphi_{1} \cdot \underset{0}{\varphi_{1}^{*}}}\mspace{14mu} {and}\mspace{14mu} {\varphi_{2} \cdot \underset{0}{\varphi_{2}^{*}}}$

of the remaining parts.

Next, intending to obtain horizontally symmetric flow, we give the boundary condition of this problem as:

$\begin{matrix} \begin{matrix} {{u_{x} = 0},{u_{y} = V}} & {{\text{:}{on}\mspace{14mu} y} = {+ b}} \\ {u_{y} = 0} & {{\text{:}{on}\mspace{14mu} y} = {- b}} \\ {u_{x} = 0} & {{\text{:}{on}\mspace{14mu} x} = 0} \end{matrix} & (1735) \end{matrix}$

The center line (x=0) is named a “boundary C_(C)”, and the boundary term of the region enclosed by C_(T), C_(B), C_(C), C_(R) is given as follows, with reference to eq. (1724):

$\begin{matrix} {0 = {{\int_{C_{T}}{\left( {{\varphi_{2} \cdot \varphi_{1}^{*}} - {\varphi_{1} \cdot \varphi_{2}^{*}}} \right)\ {c}}} - {\int_{C_{B}}{\left( {{\varphi_{2} \cdot \varphi_{1}^{*}} - {\varphi_{1} \cdot \varphi_{2}^{*}}} \right)\ {c}}} - {\int_{C_{L}}{\left( {{\varphi_{1} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \varphi_{2}^{*}}} \right)\ {c}}} + {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \varphi_{2}^{*}}} \right)\ {c}}}}} & (1736) \end{matrix}$

[Formula 714]

In order to cause the homogeneous boundary condition of eq. (1735) to be reflected on this equation, the sign of zero is described under the four primal eigenfunctions. Then, we obtain:

$\begin{matrix} {0 = {{\int_{C_{T}}{\left( {{\underset{0}{\varphi_{2}} \cdot \varphi_{1}^{*}} - {\underset{0}{\varphi_{1}} \cdot \varphi_{2}^{*}}} \right)\ {c}}} - {\int_{C_{B}}{\left( {{\underset{0}{\varphi_{2}} \cdot \varphi_{1}^{*}} - {\varphi_{1} \cdot \varphi_{2}^{*}}} \right)\ {c}}} - {\int_{C_{L}}{\left( {{\underset{0}{\varphi_{1}} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \varphi_{2}^{*}}} \right)\ {c}}} + {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \varphi_{2}^{*}}} \right)\ {c}}}}} & (1737) \end{matrix}$

As to the term in which the primal eigenfunction does not have the sign of zero, no boundary condition is designated, and the value on the boundary is unknown. An adjoint boundary condition is decided so that integration value of each of the boundaries C_(T), C_(B), C_(C), C_(R) is zero, and we obtain:

$\begin{matrix} {0 = {{\int_{C_{T}}{\left( {{\underset{0}{\varphi_{2}} \cdot \varphi_{1}^{*}} - {\underset{0}{\varphi_{1}} \cdot \varphi_{2}^{*}}} \right)\ {c}}} - {\int_{C_{B}}{\left( {{\underset{0}{\varphi_{2}} \cdot \varphi_{1}^{*}} - {\varphi_{1} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}} - {\int_{C_{C}}{\left( {{\underset{0}{\varphi_{1}} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}} + {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \underset{0}{\varphi_{1}^{*}}} + {\varphi_{2} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}}}} & (1738) \end{matrix}$

Thus, we obtain the following boundary condition imposed on the eigenfunction:

$\begin{matrix} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{T}},\mspace{31mu} \left\{ {\begin{matrix} {\varphi_{2} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{B}\left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{C}},\mspace{31mu} \left\{ {\begin{matrix} {\varphi_{1}^{*} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{R}} \right.} \right.} \right.} \right. & (1739) \end{matrix}$

If the solution functions F_(SS1), F_(AA1), F_(SA1), F_(AS1) and F_(SS2), F_(AA2), F_(SA2), F_(AS2) described in Section 11.5.11 are combined and a solution satisfying the condition of eq. (1739) is obtained, it is the eigenfunction. However, unfortunately, an eigenfunction cannot be obtained, either, in this case. An obstacle in obtaining an eigenfunction is the following in the condition of eq. (1739):

$\begin{matrix} \left\{ {\begin{matrix} {\varphi_{1}^{*} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{R}} \right. & (1740) \end{matrix}$

Therefore, as to the integration on the C_(R) of eq. (1737), the condition is relaxed in place of a strong condition as that of this equation, so as to satisfy the following in the state of integration:

[Formula 715]

0=∫_(C) _(R) (φ₁·φ₁*+φ₂·φ₂*)dc  (1741)

In Section 11.6.6, satisfying the following eauation is not intended:

$\begin{matrix} {{(*}{{)\mspace{14mu} 0} = {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \underset{0}{\varphi_{1}^{*}}} + {\varphi_{2} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}}}} & (1742) \end{matrix}$

An eigenfunction is obtained under the relaxed condition of eq. (1741), and the result is found to coincide with an eigenfunction obtained by extracting a part of eigenfunctions satisfying the following conditions and mixing the same:

$\begin{matrix} {{(2)\mspace{14mu} 0} = {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \underset{0}{\varphi_{1}^{*}}} + {\underset{0}{\varphi_{2}} \cdot \varphi_{2}^{*}}} \right)\ {c}}}} & (1743) \\ {{(3)\mspace{14mu} 0} = {\int_{C_{R}}{\left( {{\underset{0}{\varphi_{1}} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \underset{0}{\varphi_{2}^{*}}}} \right)\ {c}}}} & (1744) \\ {{(4)\mspace{14mu} 0} = {\int_{C_{R}}{\left( {{\underset{0}{\varphi_{1}} \cdot \varphi_{1}^{*}} + {\underset{0}{\varphi_{2}} \cdot \varphi_{2}^{*}}} \right)\ {c}}}} & (1745) \end{matrix}$

It is considered that since

φ_( ₀2) ⋅ φ₂^(*)

are included in (2) and (4) and

φ_( ₀1) ⋅ φ₁^(*)

are included in (3) and (4), these are canceled successfully as the number of applied modes increases, and the effect of (*) is exhibited by the

φ₁ ⋅ φ_( ₀1)^(*)

and

φ₂ ⋅ φ_( ₀2)^(*)

of the remaining parts.

Eigenfunctions present in the foregoing (2), (3), (4) are extracted, mixed, and used, whereby the effect achieved as if an eigenfunction of (*) is used is achieved. Therefore, this solution method is called “mixed-type eigenfunction method”.

As a result of that the condition on C_(R) of eq. (1739) is relaxed by eq. (1741), the boundary condition imposed on the eigenfunction is finally given as:

$\begin{matrix} {(5)\mspace{14mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{T}},\mspace{31mu} \left\{ {\begin{matrix} {\varphi_{2} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{B}\mspace{40mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{C}},\mspace{31mu} {0 = {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \varphi_{2}^{*}}} \right)\ {c}}}}}\mspace{31mu} \right.} \right.} \right.} & (1746) \end{matrix}$

[Formula 716] 11.6.2 Rules for Eigenfunction Names

In the eigenfunction shown in the subsequent section, the eigenvalues α,β are formed with the combinations given as:

$\begin{matrix} {{\alpha = \left\{ {{\frac{{2\; m} - 1}{2\; a}\pi},{\frac{2\; m}{2\; a}\pi}} \right\}}{\beta = \left\{ {{\frac{{2\; n} - 1}{2\; b}\pi},{\frac{2\; n}{2\; b}\pi},{\frac{{4\; n} - 3}{4\; b}\pi},{\frac{{4\; n} - 1}{4\; b}\pi}} \right\}}} & (1747) \end{matrix}$

where m,n are integers. There are two kinds of the eigenvalue α, and four kinds of the eigenvalue β. For example, a function F formed with the eigenvalues of

$\begin{matrix} {{\alpha = \left. {\frac{{2\; m} - 1}{2\; a}\pi}\mspace{31mu}\Leftarrow(1) \right.}{\beta = \left. {\frac{{4\; n} - 3}{4\; b}\pi}\mspace{31mu}\Leftarrow(3) \right.}} & (1748) \end{matrix}$

is represented by F₁₃ which means that α of the type (1) and β of the type (3) are used. Here, the first index is the number indicative of the type of α, and the second index is the number indicative of the type of β.

Further, this is followed by an alphabet indicating the mode form (Section 11.4.20) exhibited by the dual eigenfunction. For example, if the function F is formed with the eigenvalue α of the type (1) and the eigenvalue β of the type (3), and is in the mode SS, the function is represented by F_(13SS). The mode ZA means superimposition of SA and AA, and the mode ZS means superimposition of SS and AS.

The boundary conditions that the eigenfunction satisfies are distinguished with numbers provided at upper right. The number at upper right is the number indicative of the boundary condition. In order to clearly show that the mode form indicated by an alphabet is not that of a primal eigenfunction but that of a dual eigenfunction, (*) is added before the number at upper right, for example, F_(13SS)*².

[Formula 717] 11.6.3 Primal Eigenfunction and Dual Eigenfunction of (2)

The solution functions F_(SS1), F_(AA1), F_(SA1), F_(AS1) and F_(SS2), F_(AA2), F_(SA2), F_(AS2) mentioned in Section 11.5.11 may be combined to obtain solutions that satisfy the following condition equations (1728):

$\begin{matrix} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{T}},\mspace{31mu} \left\{ {\begin{matrix} {\varphi_{2} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{B}\left\{ {{\begin{matrix} {\varphi_{1}^{*} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{L}},\mspace{31mu} \left\{ {\begin{matrix} {\varphi_{1}^{*} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{R}} \right.} \right.} \right.} \right. & \underset{({Aforementioned})}{(1728)} \end{matrix}$

This solutions are eigenfunctions, and the following eight eigenfunctions are obtained.

Here, the functions are named according to the rules described in Section 11.6.2, and the boundary condition number is (2). λ is an eigenvalue in the primal simultaneous differential equations (1662), (1663) and the dual simultaneous differential equations (1664), (1665).

$\begin{matrix} {{(1)\mspace{14mu} F_{11\; {AA}}^{*2}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{{2\; n} - 1}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {{- \cos}\; \alpha \; x\; \cos \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; x\; \cos \; \beta \; y}} \end{matrix} \right.} & (1749) \\ {{(2)\mspace{14mu} F_{21\; {AS}}^{*2}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{{2\; n} - 1}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {\sin \; \alpha \; x\; \cos \; \beta \; y}}} \\ {\varphi_{x}^{*} = {{- \frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}}\sin \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \cos \; \beta \; y}} \end{matrix} \right.} & (1750) \\ {{(3)\mspace{14mu} F_{12\; {SA}}^{*2}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{2\; n}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {\cos \; \alpha \; x\; \sin \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; x\; \sin \; \beta \; y}} \end{matrix} \right.} & (1751) \\ \left\lbrack {{Formula}\mspace{14mu} 718} \right\rbrack & \; \\ {{(4)\mspace{14mu} F_{22\; {SS}}^{*2}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{2\; n}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {\sin \; \alpha \; x\; \sin \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \sin \; \beta \; y}} \end{matrix} \right.} & (1752) \\ {{(5)\mspace{14mu} F_{13\; {ZA}}^{*2}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 3}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\sin \; \alpha \; x\; \left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1753) \\ {{(6)\mspace{14mu} F_{23\; {ZS}}^{*2}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 3}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\cos \; \alpha \; x\; \left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1754) \\ \left\lbrack {{Formula}\mspace{14mu} 719} \right\rbrack & \; \\ {{(7)\mspace{14mu} F_{14\; {ZA}}^{*2}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 1}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\sin \; \alpha \; x\; \left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1755) \\ {{(8)\mspace{14mu} F_{24\; {ZS}}^{*2}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 1}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {{- \cos}\; \alpha \; x\; \left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1756) \end{matrix}$

These eigenfunctions have orthogonality.

11.6.4 Primal Eigenfunction and Dual Eigenfunction of (3)

The solution functions F_(SS1), F_(AA1), F_(SA1), F_(AS1) and F_(SS2), F_(AA2), F_(SA2), F_(AS2) mentioned in Section 11.5.11 may be combined to obtain solutions that satisfy the following condition equation (1729):

$\begin{matrix} {(3)\mspace{14mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{T}},\mspace{31mu} \left\{ {\begin{matrix} {\varphi_{2} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{B}\mspace{40mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{L}},\mspace{31mu} \left\{ {\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{R}} \right.} \right.} \right.} \right.} & \underset{({Aforementioned})}{(1729)} \end{matrix}$

Thes solutions are eigenfunctions, and the following eight eigenfunctions are obtained.

Here, the functions are named according to the rules described in Section 11.6.2, and the boundary condition number is (3). λ is an eigenvalue in the primal simultaneous differential equations (1662), (1663) and the dual simultaneous differential equations (1664), (1665).

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 720} \right\rbrack & \; \\ {{(1)\mspace{14mu} F_{11\; {AS}}^{*3}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{{2\; n} - 1}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {\sin \; \alpha \; x\; \cos \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \cos \; \beta \; y}} \end{matrix} \right.} & (1757) \\ {{(2)\mspace{14mu} F_{21\; {AA}}^{*3}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{{2\; n} - 1}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {{- \cos}\; \alpha \; x\; \cos \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; x\; \cos \; \beta \; y}} \end{matrix} \right.} & (1758) \\ {{(3)\mspace{14mu} F_{12\; {SS}}^{*3}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{2\; n}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {\sin \; \alpha \; x\; \sin \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \sin \; \beta \; y}} \end{matrix} \right.} & (1759) \\ {{(4)\mspace{14mu} F_{22\; {SA}}^{*3}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{2\; n}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {\cos \; \alpha \; x\; \sin \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}^{*} = {{- \frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}}\sin \; \alpha \; x\; \sin \; \beta \; y}} \end{matrix} \right.} & (1760) \\ \left\lbrack {{Formula}\mspace{14mu} 721} \right\rbrack & \; \\ {{(5)\mspace{14mu} F_{13\; {ZS}}^{*3}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 3}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\cos \; \alpha \; x\; \left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1761) \\ {{(6)\mspace{14mu} F_{23\; {ZA}}^{*3}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 3}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\sin \; \alpha \; x\; \left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1762) \\ {{(7)\mspace{14mu} F_{14\; {ZS}}^{*3}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 1}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {{- \cos}\; \alpha \; x\; \left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1763) \\ {{(8)\mspace{14mu} F_{24\; {ZA}}^{*3}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 1}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\sin \; \alpha \; x\; \left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1764) \end{matrix}$

These eigenfunctions have orthogonality.

11.6.5 Primal Eigenfunction and Dual Eigenfunction of (4) [Formula 722]

The solution functions F_(SS1), F_(AA1), F_(SA1), F_(AS1) and F_(SS2), F_(AA2), F_(SA2), F_(AS2) mentioned in Section 11.5.11 may be combined to obtain solutions that satisfy the following condition equations (1730):

$\begin{matrix} {(4)\mspace{14mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{T}},\mspace{31mu} \left\{ {\begin{matrix} {\varphi_{2} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{B}\mspace{40mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{L}},\mspace{31mu} \left\{ {\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{R}} \right.} \right.} \right.} \right.} & \underset{({Aforementioned})}{(1730)} \end{matrix}$

Thes solutions are eigenfunctions, and the following eight eigenfunctions are obtained.

Here, the functions are named according to the rules described in Section 11.6.2, and the boundary condition number is (4). λ is an eigenvalue in the primal simultaneous differential equations (1662), (1663) and the dual simultaneous differential equations (1664), (1665).

Four of the same coincide with the eigenfunctions of the second boundary condition, and four of the same coincide with the eigenfunctions of the third boundary condition. Details are shown in Table 1 in Section 11.6.6.

$\begin{matrix} {{(1)\mspace{20mu} \overset{\overset{F_{11\; {AA}}^{*2}}{\Pi}}{F_{11\; {AA}}^{*4}}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{{2\; n} - 1}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {{- \cos}\; \alpha \; x\; \cos \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; x\; \cos \; \beta \; y}} \end{matrix} \right.} & (1765) \\ {{(2)\mspace{20mu} \overset{\overset{F_{11\; {AA}}^{*2}}{\Pi}}{F_{21\; {AS}}^{*4}}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{{2\; n} - 1}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {\sin \; \alpha \; x\; \cos \; \beta \; y}}} \\ {\varphi_{x}^{*} = {{- \frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}}\sin \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \cos \; \beta \; y}} \end{matrix} \right.} & (1766) \\ \left\lbrack {{Formula}\mspace{14mu} 723} \right\rbrack & \; \\ {{(3)\mspace{14mu} \overset{\overset{F_{12\; {SA}}^{*2}}{\Pi}}{F_{12\; {SA}}^{*4}}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{2\; n}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {\cos \; \alpha \; x\; \sin \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}^{*} = {{- \frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}}\sin \; \alpha \; x\; \sin \; \beta \; y}} \end{matrix} \right.} & (1767) \\ {{(4)\mspace{14mu} \overset{\overset{F_{22\; {SS}}^{*2}}{\Pi}}{F_{22\; {SS}}^{*4}}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{2\; n}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},\mspace{14mu} {\varphi_{y} = {\sin \; \alpha \; x\; \sin \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \sin \; \beta \; y}} \end{matrix} \right.} & (1768) \\ {{(5)\mspace{14mu} \overset{\overset{F_{13\; {ZS}}^{*3}}{\Pi}}{F_{13\; {ZS}}^{*4}}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 3}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\cos \; \alpha \; x\; \left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {{- \frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1769) \\ {{(6)\mspace{20mu} \overset{\overset{F_{23\; {ZA}}^{*3}}{\Pi}}{F_{23\; {ZA}}^{*4}}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 3}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\sin \; \alpha \; x\; \left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1770) \\ \left\lbrack {{Formula}\mspace{14mu} 724} \right\rbrack & \; \\ {{(7)\mspace{14mu} \overset{\overset{F_{14\; {ZS}}^{*3}}{\Pi}}{F_{14\; {ZS}}^{*4}}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 1}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {{- \cos}\; \alpha \; x\; \left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1771) \\ {{(8)\mspace{14mu} \overset{\overset{F_{24\; {ZA}}^{*3}}{\Pi}}{F_{24\; {ZA}}^{*4}}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 1}{4\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\sin \; \alpha \; x\; \left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}},\mspace{14mu} {\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {{- \frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.} & (1772) \end{matrix}$

These eigenfunctions have orthogonality.

11.6.6 Primal Eigenfunction and Dual Eigenfunction of (5)

The solution functions F_(SS1), F_(AA1), F_(SA1), F_(AS1) and F_(SS2), F_(AA2), F_(SA2), F_(AS2) mentioned in Section 11.5.11 may be combined to obtain a solution that satisfies the following condition equation (1746):

$\begin{matrix} {(5)\mspace{14mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{T}},\mspace{31mu} \left\{ {\begin{matrix} {\varphi_{2} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{B}\mspace{40mu} \left\{ {{\begin{matrix} {\varphi_{1} = 0} \\ {\varphi_{2}^{*} = 0} \end{matrix}\mspace{14mu} {on}\mspace{14mu} C_{C}},{0 = {\int_{C_{R}}{\left( {{\varphi_{1} \cdot \varphi_{1}^{*}} + {\varphi_{2} \cdot \varphi_{2}^{*}}} \right)\ {c}}}}} \right.} \right.} \right.} & \underset{({Aforementioned})}{(1746)} \end{matrix}$

These solutions are mixed-type eigenfunctions, and the following eight eigenfunctions are obtained.

Here, the functions are named according to the rules described in Section 11.6.2, and the boundary condition number is (5). λ is an eigenvalue in the primal simultaneous differential equations (1662), (1663) and the dual simultaneous differential equations (1664), (1665).

$\begin{matrix} {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 725} \right\rbrack} & \; \\ {{(1)\mspace{14mu} \begin{matrix} {\prod\limits^{F_{11\; {AA}}^{*2}}\;} \\ F_{11\; {AA}}^{*5} \\ \prod \\ F_{11\; {AA}}^{*4} \end{matrix}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{{2\; n} - 1}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},{\varphi_{y} = {{- \cos}\; \alpha \; x\; \cos \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; x\; \cos \; \beta \; y}} \end{matrix} \right.} & (1773) \\ {\mspace{79mu} {{(2)\mspace{14mu} \begin{matrix} F_{21\; {AA}}^{*3} \\ \prod \\ F_{21\; {AA}}^{*5} \end{matrix}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{{2\; n} - 1}{2b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},{\varphi_{y} = {{- \cos}\; \alpha \; x\; \cos \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \sin \; \beta \; y}} \\ {\varphi_{y}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; x\; \cos \; \beta \; y}} \end{matrix}\mspace{14mu} \right.}} & (1774) \\ {\mspace{79mu} {{(3)\mspace{14mu} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} F_{12\; {SA}}^{*2} \\ \prod \end{matrix} \\ F_{12\; {SA}}^{*5} \end{matrix} \\ \prod \end{matrix} \\ F_{12\; {SA}}^{*4} \end{matrix}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{2\; n}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},{\varphi_{y} = {\cos \; \alpha \; x\; \sin \; \beta_{y}}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}^{*} = {{- \frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}}\sin \; \alpha \; x\; \sin \; \beta \; y}} \end{matrix} \right.}} & (1775) \\ {\mspace{79mu} {{(4)\mspace{14mu} \begin{matrix} \begin{matrix} F_{22\; {SA}}^{*3} \\ \prod \end{matrix} \\ F_{22\; {SA}}^{*5} \end{matrix}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2\; m}{2\; a}\pi}},{\beta = {\frac{2\; n}{2\; b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = 0},{\varphi_{y} = {\cos \; \alpha \; x\; \sin \; \beta \; y}}} \\ {\varphi_{x}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; x\; \cos \; \beta \; y}} \\ {\varphi_{y}^{*} = {{- \frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}}\sin \; \alpha \; x\; \sin \; \beta \; y}} \end{matrix} \right.}} & (1776) \\ {\mspace{79mu} \left\lbrack {{Formula}\mspace{14mu} 726} \right\rbrack} & \; \\ {\mspace{79mu} {{(5)\mspace{14mu} \begin{matrix} \begin{matrix} F_{13\; {ZA}}^{*2} \\ \prod \end{matrix} \\ F_{13\; {ZA}}^{*5} \end{matrix}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2\; m} - 1}{2\; a}\pi}},{\beta = {\frac{{4\; n} - 3}{4b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}},{\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.}} & (1777) \\ {\mspace{79mu} {{(6)\mspace{14mu} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} F_{23\; {ZA}}^{*3} \\ \prod \end{matrix} \\ F_{23\; {ZA}}^{*5} \end{matrix} \\ \prod \end{matrix} \\ F_{23\; {ZA}}^{*4} \end{matrix}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2m}{2a}\pi}},{\beta = {\frac{{4n} - 3}{4b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}},{\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {\frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.}} & (1778) \\ {\mspace{79mu} {{(7)\mspace{14mu} \begin{matrix} \begin{matrix} F_{14\; {ZA}}^{*2} \\ \prod \end{matrix} \\ F_{14\; {ZA}}^{*5} \end{matrix}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{{2m} - 1}{2a}\pi}},{\beta = {\frac{{4n} - 1}{4b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}},{\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {{- \frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.}} & (1779) \\ {\mspace{79mu} {{(8)\mspace{14mu} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} F_{24\; {ZA}}^{*3} \\ \prod \end{matrix} \\ F_{24\; {ZA}}^{*5} \end{matrix} \\ \prod \end{matrix} \\ F_{24\; {ZA}}^{*4} \end{matrix}} \equiv \left\{ \begin{matrix} {{\alpha = {\frac{2m}{2a}\pi}},{\beta = {\frac{{4n} - 1}{4b}\pi}},{\lambda = {R\sqrt{\alpha^{2} + \beta^{2}}}}} \\ {{\varphi_{x} = {\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}},{\varphi_{y} = 0}} \\ {\varphi_{x}^{*} = {\frac{\alpha}{\sqrt{\alpha^{2} + \beta^{2}}}\cos \; \alpha \; {x\left( {{\cos \; \beta \; y} + {\sin \; \beta \; y}} \right)}}} \\ {\varphi_{y}^{*} = {{- \frac{\beta}{\sqrt{\alpha^{2} + \beta^{2}}}}\sin \; \alpha \; {x\left( {{\cos \; \beta \; y} - {\sin \; \beta \; y}} \right)}}} \end{matrix} \right.}} & (1780) \end{matrix}$

[Formula 727]

A list of the respective equation numbers of the eigenfunction of the boundary conditions is shown in Table 1. The eigenfunctions of the second boundary condition and those of the third boundary conditions do not overlap, and the union of the two represents all of the eigenfunctions. The eigenfunctions of the fourth boundary condition and those of the fifth boundary condition can be considered to be mixtures of some extracted from the eigenfunctions of the second and third boundary conditions. The eigenfunctions of the fifth boundary condition include elements of all of the modes AA, SA, and ZA. This function group is referred to as mixed-type eigenfunctions. This function group is considered a complete system, and its orthogonality is complicated. Since F_(11AA)*⁵, F_(12SA)*⁵, F_(13ZA)*⁵, F_(14ZA)*⁵ as the first group are included in the eigenfunctions of the second boundary condition, mutual orthogonality is provided. Since F_(21AA)*⁵, F_(22SA)*⁵, F_(23ZA)*⁵, F_(24ZA)*⁵ as the second group are included in the eigenfunctions of the third boundary condition, mutual orthogonality is provided. As F_(11AA)*⁵, F_(12SA)*⁵, F_(23ZA)*⁵, F_(24ZA)*⁵ are included in the eigenfunctions of the fourth boundary condition, mutual orthogonality is provided. In an inner product of the first group and the second group, the functions of the mode AA and the functions of the mode SA are orthogonal to each other. As a special example, F_(13ZA)*⁵ and F_(24ZA)*⁵ are orthogonal, and F_(14ZA)*⁵ and F_(23ZA)*⁵ are orthogonal as well.

TABLE 1 Equation Number of Eigenfunction Boundary Name Condition Number α, β Mode 2 3 4 5 11 AA (1749) (1765) (1773) AS (1757) 21 AA (1758) (1774) AS (1750) (1766) 12 SA (1751) (1767) (1775) SS (1759) 22 SA (1760) (1776) SS (1752) (1768) 13 ZA (1753) (1777) ZS (1761) (1769) 23 ZA (1762) (1770) (1778) ZS (1754) 14 ZA (1755) (1779) ZS (1763) (1771) 24 ZA (1764) (1772) (1780) ZS (1756)

[Formula 728]

States of No. 1 mode of the primal eigenfunctions and dual eigenfunctions in the cases of (1) to (8) under the fifth boundary condition are shown in FIGS. 60 to 67.

[Formula 732] 11.6.7 Mixed-Type Eigenfunction Method

There are eight mixed-type eigenfunctions of the fifth boundary condition with respect to combinations of integers m,n. These are provided with serial numbers, and let the serial number be k. Let the k-th eigenvalue be λ_(k), let the k-th primal eigenfunction be φ_(jk), and let the k-th dual eigenfunction be φ_(jk)*. Similarly to Section 11.5.12, the simultaneous partial differential equations (1548), (1549) are in the form of eq. (23) as follows:

$\begin{matrix} {{\sum\limits_{j}{L_{ij}u_{j}}} = f_{i}} & \begin{matrix} {\mspace{140mu} (23)} \\ {({Aforementioned}\;)\;} \end{matrix} \end{matrix}$

An index B is added to a term that satisfies an inhomogeneous boundary condition so as to let the term be u_(Bj), and an index H is added to a term that satisfies a homogeneous boundary condition so as to let the term be u_(Hj). A primal velocity is expressed by a sum of these, which is given as:

u _(j) ≡u _(Bj) +u _(Hj)  (24) (Aforementioned)

Substituting this equation (24) into eq. (23), we obtain the following simultaneous partial differential equation (40) represented by a homogeneous boundary condition:

$\begin{matrix} {{\sum\limits_{j}{L_{ij}u_{Hj}}} = f_{Hi}} & \begin{matrix} {\mspace{140mu} (40)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

where the external force term f_(H) satisfies:

$\begin{matrix} {f_{Hi} \equiv {f_{i} - {\sum\limits_{j}{L_{ij}u_{Bj}}}}} & \begin{matrix} {\mspace{140mu} (41)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

Here, the boundary function differential term f_(Bi) is defined as:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 733} \right\rbrack & \; \\ {f_{Bi} \equiv {\sum\limits_{j}{L_{ij}u_{Bj}}}} & \begin{matrix} {\mspace{124mu} (1696)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

Then, we obtain:

f _(Hi) ≡f _(i) −f _(Bi)  (1697) (Aforementioned)

The solution function u_(Hj) is expressed by a sum of mixed-type eigenfunctions as follows:

$\begin{matrix} {u_{Hj} \equiv {\sum\limits_{k}{c_{k}\varphi_{jk}}}} & \begin{matrix} {\mspace{140mu} (69)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

Substituting eq. (69) into the differential equation (40) gives:

$\begin{matrix} {{\sum\limits_{k}{c_{k}\lambda_{k}w_{i}\varphi_{ik}^{*}}} = f_{Hi}} & \begin{matrix} {\mspace{135mu} (109)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

Let the number of applied modes be K, and we obtain an inner product of this equation with the p-th dual eigenfunction φ_(ip)*, which is as follows:

$\begin{matrix} {{\sum\limits_{k}^{K}{c_{k}\lambda_{k}{\sum\limits_{i}{\int_{S}{w_{i}{\varphi_{ik}^{*} \cdot \varphi_{ip}^{*}}\ {s}}}}}} = {\sum\limits_{i}{\int_{S}{{f_{Hi} \cdot \varphi_{ip}^{*}}\ {s}}}}} & (1781) \end{matrix}$

The mixed-type eigenfunctions, which have complicated orthogonality, do not have such convenience that the summing sign of k can be removed and the coefficient c_(k) is determined immediately as is the case with eq. (110). When p is varied from 1 to K in eq. (1781), a matrix format is obtained. Let the inner products be given as:

$\begin{matrix} {s_{p\; k} \equiv {\lambda_{k}{\sum\limits_{i}{\int_{S}{w_{i}{\varphi_{ik}^{*} \cdot \varphi_{ip}^{*}}\ {s}}}}}} & (1782) \\ {r_{p} \equiv {\sum\limits_{i}{\int_{S}{{f_{Hi} \cdot \varphi_{ip}^{*}}\ {s}}}}} & (1783) \end{matrix}$

Then, we obtain:

$\begin{matrix} {{\begin{bmatrix} s_{11} & s_{12} & \Lambda & s_{1\; K} \\ s_{21} & s_{22} & \Lambda & s_{2\; K} \\ M & M & O & M \\ s_{K\; 1} & S_{K\; 2} & \Lambda & s_{KK} \end{bmatrix}\begin{Bmatrix} c_{1} \\ c_{2} \\ M \\ c_{K} \end{Bmatrix}} = \begin{Bmatrix} r_{1} \\ r_{2} \\ M \\ r_{K} \end{Bmatrix}} & (1784) \end{matrix}$

By solving this simultaneous linear equation, the coefficient c_(k) is settled. This method is referred to as a “mixed-type eigenfunction method”. The c_(k) thus settled is returned to eq. (69), and the solution function u_(Hj) is obtained. When u_(Hj) is returned to eq. (24), the velocity u_(j) is obtained. On the other hand, when it is returned to eq. (109), the external force term f_(H) is reconstructed, which is useful for verification. In other words, the solution method (1784) is an eigenfunction method in the Hilbert space in which the velocity is expressed with the primal eigenfunction φ_(j), and the external force term f_(H) is expressed with the dual eigenfunction φ_(i)*.

[Formula 734]

The following description shows that the procedure of taking an inner product with the dual eigenfunction φ_(ip)* of eq. (1781), that is, the mixed-type eigenfunction method, is also equivalent to the least-squares method.

Similarly to Section 5.2, the primal variation δu_(i) and the dual variation δu_(i)* are given as:

$\begin{matrix} {{\delta \; u_{i}} = {{\delta \; u_{Hi}} \equiv {\sum\limits_{k}{\delta \; c_{k}\varphi_{ik}}}}} & \begin{matrix} {\mspace{135mu} (120)} \\ ({Aforementioned}\;) \end{matrix} \\ {{\delta \; u_{i}^{*}} = {{\delta \; u_{Hi}^{*}} \equiv {\sum\limits_{k}{\delta \; c_{k}^{*}\varphi_{ik}^{*}}}}} & \begin{matrix} {\mspace{135mu} (121)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

Therefore, combining a plurality of procedures for taking an inner product with the function φ_(i)*, which is a technique used in the process of obtaining eq. (1781), is equivalent to taking an inner product with the dual variation δu_(i)* of eq. (121). The inner product of eq. (1781) is basically an inner product of eq. (40) and the function φ_(i)*. Transforming eq. (40) gives:

$\begin{matrix} {{{\sum\limits_{j}{L_{ij}u_{Hj}}} - f_{Hi}} = 0} & (1785) \end{matrix}$

Taking an inner product thereof with the dual variation δu_(i)* gives:

$\begin{matrix} {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{Hj}}} - f_{Hi}} \right) \cdot \delta}\; u_{i}^{*}\ {s}}}} = 0} & (1786) \end{matrix}$

Substituting the external force term f_(H) of eq. (41) into this equation gives:

$\begin{matrix} {{\sum\limits_{i}{\int_{S}{{\left\{ {{\sum\limits_{j}{L_{ij}\left( {u_{Bj} + u_{Hj}} \right)}} - f_{i}} \right\} \cdot \delta}\; u_{i}^{*}\ {s}}}} = 0} & (1787) \end{matrix}$

Expressing the velocity of this equation with the primal velocity u_(j) of eq. (24) gives:

$\begin{matrix} {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}^{*}\ {s}}}} = 0} & \begin{matrix} {\mspace{135mu} (122)} \\ ({Aforementioned}\;) \end{matrix} \end{matrix}$

On the other hand, according to eqs. (105), (120), we obtain the following, regarding a sum of differential coefficients of the primal variation δu_(j):

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 735} \right\rbrack & \mspace{14mu} \\ {{\sum\limits_{j}{L_{ij}\delta \; u_{j}}} = {{\sum\limits_{k}{\delta \; c_{k}{\sum\limits_{j}{L_{ij}\varphi_{jk}}}}} = {\sum\limits_{k}{\delta \; c_{k}\lambda_{k}w_{i}{\varphi_{ik}^{*}({Aforementioned})}}}}} & (124) \end{matrix}$

In the primal and dual eigenfunctions in which the same weight w is used for every component, the following is defined:

δc _(k) *≡δc _(k)λ_(k) w _(i) , w _(i) =w  (125) (Aforementioned)

Therefore, according to eqs. (121), (124), (125), we can recognize the dual variation δu_(i)* as:

$\begin{matrix} {{\delta \; u_{j}^{*}} = {{\sum\limits_{j}{L_{ij}\delta \; u_{j}}} = {\delta {\sum\limits_{j}{L_{ij}{u_{j}({Aforementioned})}}}}}} & (126) \end{matrix}$

Substituting this equation to eq. (122) gives:

$\begin{matrix} {{\sum\limits_{i}{\int_{S}{{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}{\sum\limits_{j}{L_{ij}u_{j}{s}}}}}} = {0({Aforementioned})}} & (127) \end{matrix}$

This equation is equivalent to a case where the functional Π is given as

$\begin{matrix} {\Pi \equiv {\sum\limits_{i}{\int_{S}{\left( {{\sum\limits_{j}{L_{ij}u_{j}}} - f_{i}} \right)^{2}{{s({Aforementioned})}}}}}} & (128) \end{matrix}$

and its variation is zero. In other words, the least-squares method is applied as the variational principle.

11.6.8 Boundary Function

As the boundary function u_(Bj) satisfying the following boundary condition of present problem

u _(x)=0, u _(y) =V :on y=+b

u _(y)=0, :on y=+b,  (1720) (Aforementioned)

the following three types of the boundary functions are defined:

$\begin{matrix} (1) & \; \\ \left\{ \begin{matrix} {u_{B\; 1} \equiv u_{Bx} \equiv 0} \\ {{u_{B\; 2} \equiv u_{By}} = {V\; \frac{y + b}{2b}}} \end{matrix} \right. & (1788) \\ (2) & \; \\ \left\{ \begin{matrix} {u_{B\; 1} \equiv u_{Bx} \equiv {{- V}\; \frac{x}{a}\left( \frac{y - b}{2b} \right)^{2}}} \\ {u_{B\; 2} \equiv u_{B\; y} \equiv {V\; \frac{y + b}{2b}}} \end{matrix} \right. & (1789) \\ (3) & \; \\ \left\{ \begin{matrix} {u_{B\; 1} \equiv u_{Bx} \equiv {V\; \sin \; \frac{x}{a}\pi \; \sin \frac{\; {y - b}}{4b}\pi}} \\ {u_{B\; 2} \equiv u_{By} \equiv {V\; \frac{y + b}{4b}\pi}} \end{matrix} \right. & (1790) \end{matrix}$

[Formula 736]

We can see that all of the same have a velocity v at the top edge as a coefficient and satisfy the boundary condition of eq. (1720).

With use of the boundary function differential term f_(Bi)

$\begin{matrix} {{f_{Bi} \equiv {\sum\limits_{j}{L_{ij}u_{Bj}}}},({Aforementioned})} & (1696) \end{matrix}$

the external force term f_(H) is given as:

f _(Hi) ≡f _(i) −f _(Bi)  (1697) (Aforementioned)

According to the equation of continuity and the vortex-free condition, the external force term f_(i) is zero, as is the case with eq. (1719). Therefore, the external force term f_(H) is given as

f _(Hi) =−f _(Bi)  (1791)

This indicates that the external force term is expressed only with the boundary function differential term f_(Bi). Rasnantive f_(Bi) is calculated as follows:

$\begin{matrix} (1) & \; \\ \left\{ \begin{matrix} {f_{B\; 1} \equiv f_{Bx} \equiv {V\; \frac{1}{2b}}} \\ {{f_{B\; 2} \equiv f_{By}} = 0} \end{matrix} \right. & (1792) \\ (2) & \; \\ \left\{ \begin{matrix} {f_{B\; 1} \equiv f_{B\; x} \equiv {V\; \frac{{2{ab}} - \left( {y - b} \right)^{2}}{4{ab}^{2}}}} \\ {f_{B\; 2} \equiv f_{{By}\;} \equiv {V\; \frac{{x\left( {y - b} \right)}^{2}}{2{ab}^{2}}}} \end{matrix} \right. & (1793) \\ (3) & \; \\ \left\{ \begin{matrix} {{f_{B\; 1} \equiv f_{B\; x}} = {\pi \; {V\left( {{\frac{1}{4b}\cos \frac{\; {y + b}}{4b}\pi} + {\frac{1}{a}\cos \; \frac{x}{a}\pi \; \sin \; \frac{y - b}{4b}\pi}} \right)}}} \\ {{f_{B\; 2} \equiv f_{By}} = {{- \pi}\; V\; \frac{1}{4b}\sin \; \frac{x}{a}\pi \; \cos \; \frac{y - b}{4b}\pi}} \end{matrix} \right. & (1794) \end{matrix}$

Focusing on the term f_(By) of (1), (3), f_(By)=0 at the bottom edge (y=−b) on the wall surface, whereas in the case of (2), f_(By)≠0.

[Formula 737]

On the other hand, all of the dual components of the mixed-type eigenfunctions satisfy φ₂*≡φ_(y)*=0 at the bottom edge. Therefore, in the case of (2), errors remain necessarily in the vortex distribution at the bottom edge. Actually, u_(Bj) is given to the three types of boundary functions and calculation is performed by the mixed-type eigenfunction method. As a result, while (1), (3) give excellent results, only (2) still has errors.

To execute an analysis that does not leave errors on the boundary, it is necessary to give a boundary function U_(Bj) suitable for a boundary value of a dual eigenfunction.

11.6.9 Calculation Example

Shown below are results obtained by using the boundary function of eq. (1788) in Section 11.6.8 (1), using the mixed-type eigenfunctions satisfying the fifth boundary condition in Section 11.6.6, and applying the mixed-type eigenfunction method in Section 11.6.7. Here, the velocity is made dimensionless by the velocity of flow V.

A state of the dimensionless velocity in a square region, with the mode number m×n of the eigenfunction being set to at most 3×3, is shown in FIG. 68.

[Formula 738]

In FIG. 68, we can see a state in which streams of fluid flowing from the left and right edges collide each other on the center line, turn upward and join, thereby becoming uniform flow at the top edge.

A state of the dimensionless velocity, with the mode number m×n of the eigenfunction being set to at most 4×4, is shown in FIG. 69.

In FIG. 69, countercurrents occur at the left and right edges. More specifically, while the fluid flows into the region in the vicinities of the bottom edge, the fluid flows out in the vicinities of the middle area. We can see that this results in that the uniform flow at the top edge is stabilized.

[Formula 739]

On the other hand, we can also see that as compared with the velocity at the upper end, the velocity of flow at the lower end increases.

A state of the dimensionless velocity, with the mode number m×n of the eigenfunction being set to at most 6×6, is shown in FIG. 70.

In FIG. 70, the countercurrents at the left and right edges are stronger. More specifically, we can see that as compared with the velocity of flow at the top edge, the inlet velocity of flow at the bottom edge and the outlet velocity of flow in the middle area increase. We can see that this results in that the uniform flow at the top edge is stabilized.

[Formula 740]

11.6.10 Comparison with Analytical Solution

In problems of hydrodynamics (hydrodynamics), the boundary element method (BEM) is used often. As in the example of Section 11.6.9, a problem such that two velocity components as boundary conditions are given to one boundary (condition for the top edge) or two velocity components are unknown (condition for the left and right edges) cannot be solved by the contemporary boundary element method.

However, it is possible to perform equivalent calculation by superimposing analytical solutions, and to verify the calculation result of Section 11.6.9.

The combinations of the solutions of harmonic equations that the velocity potential φ should satisfy are given as:

$\begin{matrix} {{{{\varphi \left( {x,y} \right)} \equiv {{X(x)} \cdot {Y(y)}}} = {\begin{pmatrix} {\sinh \; {vx}} \\ {\cosh \; {vx}} \end{pmatrix} \times \begin{pmatrix} {\sin \; {vy}} \\ {\cos \; {vy}} \end{pmatrix}}}({Aforementioned})} & (1571) \end{matrix}$

From these combinations, two functions that satisfy the following boundary conditions are obtained:

$\begin{matrix} {{{u_{x} \equiv \frac{\partial\varphi}{\partial x}} = {{0\text{:}\mspace{14mu} {on}\mspace{14mu} y} = {+ b}}}{{u_{y} \equiv \frac{\partial\varphi}{\partial y}} = {{0\text{:}\mspace{14mu} {on}\mspace{14mu} y} = {- b}}}} & (1795) \end{matrix}$

Let n be an integer, and define the value of v as:

$\begin{matrix} {v_{n} = {\frac{{2n} - 1}{4b}\pi}} & (1796) \end{matrix}$

Then, we obtain the foregoing two functions as follows:

φ=cos hv _(n) x cos v _(n)(y+b)  (1797)

φ=sin hv _(n) x cos v _(n)(y+b)  (1798)

Among these two equations, eq. (1797), which represents horizontally symmetric flow, is used. Superimposing the equations (1797) with respect to various integers n gives the velocity potential Φ for the whole as follows:

$\begin{matrix} {\Phi \equiv {\sum\limits_{n}{A_{n}\cosh \; v_{n}x\; \cos \; {v_{n}\left( {y + b} \right)}}}} & (1799) \end{matrix}$

Here, A_(n) represents a coefficient. The velocity in the y direction indicated by the velocity potential Φ of this equation is given as:

$\begin{matrix} {{u_{y} \equiv \frac{\partial\Phi}{\partial y}} = {- {\sum\limits_{n}{A_{n}v_{n}\cosh \; v_{n}x\; \sin \; {v_{n}\left( {y + b} \right)}}}}} & (1800) \end{matrix}$

[Formula 741]

In order to satisfy the boundary condition in which the velocity is V at the top edge, y=b is given to this equation, and the equation is transformed to:

$\begin{matrix} {V = {- {\sum\limits_{n}{A_{n}v_{n}\cosh \; v_{n}x\; \sin \mspace{11mu} 2v_{n}b}}}} & (1801) \end{matrix}$

According to eq. (1796), this equation is further transformed to:

$\begin{matrix} {V = {\sum\limits_{n}{A_{n}\frac{{2n} - 1}{4b}{\pi \left( {- 1} \right)}^{n}\cosh \; v_{n}x}}} & (1802) \end{matrix}$

Let the coefficient A_(n) satisfy:

A _(n) ≡c _(n) bV  (1803)

Then, making the equation (1802) dimensionless gives:

$\begin{matrix} {1 = {\sum\limits_{n}{c_{n}\frac{{2n} - 1}{4}{\pi \left( {- 1} \right)}^{n}\cosh \; v_{n}x}}} & (1804) \end{matrix}$

It is possible to multiply this equation by cos hv_(n)x with respect to various integers n, and integrating the same so as to create a simultaneous linear equation, thereby deciding the coefficient c_(n). This is a technique generally called “the method of weighted residual”. Further, this is also the least-squares method having cos h v_(n)x as a basis function.

With use of the coefficient c_(n) decided in this way, the whole velocity potential Φ is given as:

$\begin{matrix} {\Phi \equiv {\sum\limits_{n}{c_{n}{bV}\; \cosh \; v_{n}x\; \cos \; {v_{n}\left( {y + b} \right)}}}} & (1805) \end{matrix}$

Velocity components are determined from this equation, and are divided by the velocity V, thereby being made dimensionless. The results are shown below.

A state of the dimensionless velocity in a region that is square, with the mode number n being set to at most 3, is shown in FIG. 71.

FIG. 68 is very similar to FIG. 71. We can see a state in which streams of fluid flowing from the left and right edges collide each other on the center line, turn upward and join, thereby becoming uniform flow at the top edge.

A state of the dimensionless velocity, with the mode number n being set to at most 4, is shown in FIG. 72

FIG. 69 is very similar to FIG. 72. Countercurrents occur at the left and right edges. More specifically, while the fluid flows into the region in the vicinities of the bottom edge, the fluid flows out in the vicinities of the middle area. We can see that this results in that the uniform flow at the top edge is stabilized. On the other hand, we can also see that as compared with the velocity of low at the top edge, the inlet velocity of flow at the bottom edge is greater.

A state of the dimensionless velocity, with the mode number n being set to at most 8, is shown in FIG. 73.

[Formula 744]

FIG. 70 is very similar to FIG. 73. The countercurrents at the left and right edges are stronger. More specifically, we can see that as compared with the velocity of flow at the top edge, the inlet velocity of flow at the bottom edge and the outlet velocity of flow in the middle area increase. We can see that this results in that the uniform flow at the top edge is stabilized.

Comparing FIGS. 70 to 73, the state of flow significantly varies with the number of modes used. Every solution is obtained by superimposing the velocity potentials satisfying the boundary condition equation (1795), thereby providing a physically feasible flow field, but differences exist in the uniformity at the top edge. How different the flow fields are regarding the uniformity of velocity distribution at the top edge is studied. Let the y-direction velocity at the upper end obtained from the velocity potential of eq. (1805) using N modes be v_(N), and an error with respect to the velocity of flow V is defined as:

$\begin{matrix} {{err} \equiv {\frac{V - V_{N}}{V} \times {100\lbrack\%\rbrack}}} & (1806) \end{matrix}$

Regarding every flow field, error distribution at the top edge thus determined is shown.

[Formula 745]

FIG. 71 shows the flow field when N=3. The error distribution err at the top edge is shown in FIG. 74.

FIG. 72 shows the flow field when N=4. The error distribution err at the top edge is shown in FIG. 75.

[Formula 746]

FIG. 73 shows a flow field when N=8. The error distribution err at the top edges is shown in FIG. 76.

Comparing FIGS. 74 to 76, we can see that as more modes are used, the velocity distribution at the top edge is closer to strictly uniform flow. The error when 3 modes are used (FIG. 74) is on the order of 10⁻¹[%], and the error when 8 modes are used (FIG. 76) is on the order of 10⁻⁵[%]. The analytical solutions indicate that the state of the flow field varies significantly as an originally slight error is made further smaller with use of a high-order mode. Conversely speaking, the state of flow-in and flow-out to and from the left and right edges is significantly different depending on whether roughly uniform flow is to be achieved or strictly uniform flow is to be achieved.

Further, the results determined by the mixed-type eigenfunction method (FIGS. 68 to 70) also indicated that as the number of used modes increases, changes similar to those described in the present section occur. These can be considered to prove the calculation results shown in Section 11.6.9 is reasonable, which shows the effectiveness of the mixed-type eigenfunction method.

[Formula 747] 11.7 Boundary Condition of Finite Element Method

This section should better be included in Section 11.3.

11.7.1 User Input of Boundary Condition

In Sections 8.11, 9.2, and 11.3.11, it is described that the nodal displacement and the nodal forces are divided into a known part and an unknown part.

The following description provides supplementary explanation about what the user should input into a program and how the user should deal with a self-adjoint problem and a non-self-adjoint problem. In usual processing used in the currently-practiced finite element method program, what a user inputs is only information of the known part of the nodal displacement and the known part of the nodal external force. This is because the object is limited to a self-adjoint problem.

In contrast, with respect to a non-self-adjoint problem, there are two processing methods.

(1) Degrees of freedom that a user does not input in accordance with the currently-practiced finite element method program are all recognized as known parts of the nodal external force, and the value of zero is given internally. Therefore, the user has to separately input degrees of freedom to be set for the unknown part of the nodal displacement and the unknown part of the nodal external force.

(2) All of degrees of freedom that the user does not input are treated as unknown parts. Therefore, the user should not omit inputting the value of zero, in the case where the value of zero is to be given to the nodal external force.

In order to make the sense of use closer to the sense of use of the currently-practiced finite element method, the method (1) is preferable.

[Formula 748]

The following describes the background of the forgoing description. The both sides of the following equation (250) showing the boundary term R is divided by G, and the order of F and U is inverted:

$\begin{matrix} {{R \equiv {\begin{Bmatrix} \underset{1 \times n}{\left\{ {F/G} \right\}^{T}} & \underset{1 \times n}{\left\{ {- U} \right\}^{T}} \end{Bmatrix}\begin{Bmatrix} \underset{n \times 1}{\left\{ U^{*} \right\}} \\ \underset{n \times 1}{\left\{ {F^{*}/G} \right\}} \end{Bmatrix}}}({Aforementioned})} & (250) \end{matrix}$

Then, we obtain:

$\begin{matrix} {\frac{R}{G} \equiv {\begin{Bmatrix} \underset{1 \times n}{\left\{ U \right\}^{T}} & \underset{1 \times n}{\left\{ F \right\}^{T}} \end{Bmatrix}\begin{Bmatrix} \underset{n \times 1}{\left\{ {- F^{*}} \right\}} \\ \underset{n \times 1}{\left\{ U^{*} \right\}} \end{Bmatrix}}} & (1807) \end{matrix}$

Dividing the nodal displacement U into a known part U_(b) and an unknown part U_(v) and rearranging the same gives:

$\begin{matrix} {\underset{n \times 1}{\left\{ U \right\}} \equiv \begin{Bmatrix} \underset{n_{U_{b}} \times 1}{\left\{ U_{b} \right\}} \\ \underset{n_{U_{v}} \times 1}{\left\{ U_{v\;} \right\}} \end{Bmatrix}} & (1808) \end{matrix}$

The dual nodal external force F* has to be rearranged in accordance with this. The dual nodal external force corresponding to the known part U_(b) is an unknown part, and let the same be F_(v)*. The dual nodal external force corresponding to the unknown part U_(v) is a known part, and let the same be F_(b)*. As a result, we obtain:

$\begin{matrix} {\underset{n \times 1}{\left\{ F^{*} \right\}} = \begin{Bmatrix} \underset{n_{F_{v}^{*}} \times 1}{\left\{ F_{v}^{*} \right\}} \\ \underset{n_{F_{b}^{*}} \times 1}{\left\{ F_{b}^{*} \right\}} \end{Bmatrix}} & (1809) \end{matrix}$

Respective numbers are as follows, as is the case with eq. (252):

n _(F) _(v) _(*) =n _(U) _(b)

n _(F) _(b) _(*) =n _(U) _(v)   (1810)

Similarly, Dividing the nodal external force F into a known part F_(b) and an unknown part F_(v) and rearranging the same gives:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 749} \right\rbrack & \; \\ {\underset{n \times 1}{\left\{ F \right\}} \equiv \begin{Bmatrix} \underset{n_{F_{b}} \times 1}{\left\{ F_{v} \right\}} \\ \underset{n_{F_{v}} \times 1}{\left\{ F_{b} \right\}} \end{Bmatrix}} & (1811) \end{matrix}$

The dual nodal displacement U* has to be rearranged in accordance with this. The dual nodal displacement corresponding to the known part F_(b) is an unknown part, and let the same be U_(v)*. The dual nodal displacement corresponding to the unknown part F_(v) is a known part, and let the same be U_(b)*. As a result, we obtain:

$\begin{matrix} {\underset{n \times 1}{\left\{ U^{*} \right\}} \equiv \begin{Bmatrix} \underset{n_{U_{v}^{*}} \times 1}{\left\{ U_{v}^{*} \right\}} \\ \underset{n_{U_{b}^{*}} \times 1}{\left\{ U_{b}^{*} \right\}} \end{Bmatrix}} & (1812) \end{matrix}$

Respective numbers are as follows, as is the case with eq. (252):

n _(U) _(v) _(*) =n _(F) _(b)

n _(U) _(b) _(*) =n _(F) _(v)   (1813)

Substituting eqs. (1808), (1809), (1811), and (1812) into eq. (1807) gives:

$\begin{matrix} {\frac{R}{G} \equiv {\begin{Bmatrix} \underset{1 \times n_{U_{b}}}{\left\{ U_{b} \right\}^{T}} & \underset{1 \times n_{U_{v\;}}}{\left\{ U_{v} \right\}^{T}} & \underset{1 \times n_{F_{b}}}{\left\{ F_{b} \right\}^{T}} & \underset{1 \times n_{F_{v}}}{\left\{ F_{v} \right\}^{T}} \end{Bmatrix}\begin{Bmatrix} \underset{n_{F_{v}^{*}} \times 1}{\left\{ {- F_{v}^{*}} \right\}} \\ \underset{n_{F_{b}^{*}} \times 1}{\left\{ {- F_{b}^{*}} \right\}} \\ \underset{n_{U_{v}^{*}} \times 1}{\left\{ U_{v}^{*} \right\}} \\ \underset{n_{U_{b}^{*}} \times 1}{\left\{ U_{b}^{*} \right\}} \end{Bmatrix}}} & (1814) \end{matrix}$

This is equivalent to eq. (251) as follows:

$\begin{matrix} \left\lbrack {{Formula}\mspace{14mu} 750} \right\rbrack & \; \\ {{R \equiv {\begin{Bmatrix} \underset{1 \times n_{F_{b}}}{\left\{ {F_{b}/G} \right\}^{T}} & \underset{1 \times n_{F_{v}}}{\left\{ {F_{v}/G} \right\}^{T}} & \underset{1 \times n_{U_{b}}}{\left\{ {- U_{b}} \right\}^{T}} & \underset{1 \times n_{U_{v\;}}}{\left\{ {- U_{v}} \right\}^{T}} \end{Bmatrix}\begin{Bmatrix} \underset{n_{U_{v}^{*}} \times 1}{\left\{ U_{v}^{*} \right\}} \\ \underset{n_{U_{b}^{*}} \times 1}{\left\{ U_{b}^{*} \right\}} \\ \underset{n_{F_{v}^{*}} \times 1}{\left\{ {F_{v}^{*}/G} \right\}} \\ \underset{n_{F_{b}^{*}} \times 1}{\left\{ {F_{b}^{*}/G} \right\}} \end{Bmatrix}}}({Aforementioned})} & (251) \end{matrix}$

If the known parts F_(b),U_(b) are replaced with zero, this means that a homogeneous boundary condition is given. Here, by a condition such that the boundary term R is zero, a homogeneous adjoint boundary condition is obtained, and F_(b)*,U_(b)* become zero. In the case where the combinations of the known parts F_(b), F_(b)* and the known parts U_(b),U_(b)* coincide, this is a self-adjoint boundary condition. In the case where they do not coincide, this is a non-self-adjoint boundary condition.

In order to clearly show the equation (1814) when the self-adjoint boundary condition is obtained, let the number of the known parts U_(b) be n₁, and let the number of known part F_(b) be n₂, and then, according to eqs. (1810) and (1813), we obtain:

n ₁ =n _(U) _(b) =n _(F) _(v) _(*) =n _(F) _(v) =n _(U) _(b) _(*)

n ₂ =n _(F) _(b) =n _(U) _(v) _(*) =n _(U) _(v) =n _(F) _(b) _(*)   (1815)

Reflecting this on eq. (1814) gives:

$\begin{matrix} {\frac{R}{G} \equiv {\begin{Bmatrix} \underset{1 \times n_{1}}{\left\{ U_{b} \right\}^{T}} & \underset{1 \times n_{2}}{\left\{ U_{v} \right\}^{T}} & \underset{1 \times n_{2}}{\left\{ F_{b} \right\}^{T}} & \underset{1 \times n_{1}}{\left\{ F_{v} \right\}^{T}} \end{Bmatrix}\begin{Bmatrix} \underset{n_{1} \times 1}{\left\{ {- F_{v}^{*}} \right\}} \\ \underset{n_{2} \times 1}{\left\{ {- F_{b}^{*}} \right\}} \\ \underset{n_{2} \times 1}{\left\{ U_{v}^{*} \right\}} \\ \underset{n_{1} \times 1}{\left\{ U_{b}^{*} \right\}} \end{Bmatrix}}} & (1816) \end{matrix}$

[Formula 751]

This equation shows that when, regarding each node, only either one of the displacement and the external force is given, the boundary term R becomes zero. This therefore makes it clear that the format of the currently-practiced finite element method is constructed on the premise of the self-adjoint boundary condition. When a user inputs a known part of a nodal displacement, this is considered to mean that the designation is completed regarding the nodal displacement, the degree of freedom of the known part U_(b) is fixed and the number n₁ is settled. At the same time, regarding the nodal external force, the degree of freedom of the unknown part F_(v) is settled. According to the other degrees of freedom, the degrees of freedom of the known part F_(b) of the nodal external force and that of the unknown part U_(v) of the nodal displacement are settled, and the number of the same is given as n₂=n−n₁. The value of F_(b) is set to zero, so that no external force is assumed to act. If there is a known part of a nodal external force input by a user, the part assumed to be zero is changed to the value thus input by the user. Through the foregoing process, information necessary for calculation is prepared.

In the case of a non-self-adjoint problem, two handling methods are applicable.

The above-described method (1) reflects an unknown part designated by a user, in addition to the above-described process of the currently-practiced finite element method. The currently-practiced finite element method does not have a function of designating an unknown part.

The above-described method (2) is configured such that all the designation of known parts is performed by a user, and an unknown part is determined as a remaining degree of freedom.

[Formula 752] 11.7.2 Determination of Self-Adjointness of Boundary Condition

In the finite element method using conventional elements, the equation of motion (434) for the whole system is recognized, with use of a unit matrix [I], as:

$\begin{matrix} {{{\underset{n \times n}{\lbrack K\rbrack}\underset{n \times 1}{\left\{ U \right\}}} = {\underset{n \times n}{\lbrack I\rbrack}\underset{n \times 1}{\left\{ F \right\}}}}({Aforementioned})} & (441) \end{matrix}$

Then, we obtain:

$\begin{matrix} {{{\begin{bmatrix} \underset{n \times n}{\lbrack K\rbrack} & \underset{n \times n}{\left\lbrack {- I} \right\rbrack} \end{bmatrix}\begin{Bmatrix} \underset{n \times 1}{\left\{ U \right\}} \\ \underset{n \times 1}{\left\{ F \right\}} \end{Bmatrix}} = \underset{n \times 1}{\left\{ 0 \right\}}}({Aforementioned})} & (442) \end{matrix}$

Combining the known part U_(b) of the nodal displacement U and the known part F_(b) of the nodal external force F, we define the nodal known part s_(b) as:

$\begin{matrix} {{\underset{n_{b} \times 1}{\left\{ s_{b} \right\}} \equiv \begin{Bmatrix} \underset{n_{U_{b}} \times 1}{\left\{ U_{b} \right\}} \\ \underset{n_{F_{b}} \times 1}{\left\{ F_{b} \right\}} \end{Bmatrix}}({Aforementioned})} & (443) \end{matrix}$

Combining the unknown part U_(v) of the nodal displacement U, and the unknown part F_(v) of the nodal external force F, we define the nodal unknown part s_(v) as:

$\begin{matrix} {{\underset{n_{v} \times 1}{\left\{ s_{v} \right\}} \equiv \begin{Bmatrix} \underset{n_{U_{v}} \times 1}{\left\{ U_{v} \right\}} \\ \underset{n_{F_{v}} \times 1}{\left\{ F_{v} \right\}} \end{Bmatrix}}({Aforementioned})} & (444) \end{matrix}$

From the knowledge of eq. (1816), in the case where degrees of freedom of U_(b) and F_(v) coincide and degrees of freedom of F_(b) and U_(v) coincide, this is a self-adjoint boundary condition. In this case, if analysis is performed by eq. (436), calculation time is shorter. Therefore, branching the calculation routine via a self-adjoint determination routine enables more efficient calculation. It should be noted that Section 9.4 shows that a plurality of solutions are obtained in the case of a non-self-adjoint boundary condition, but in the case where a unique solution is obtained, there is no necessity to apply the least-squares method.

12. End 12.1 End 12.1.1 End

End 

1-16. (canceled)
 17. An information processing device comprising: an initial equation decision unit that reads data indicating a structure of a system as an object of processing and properties of a constituent element of the system, and decides n initial equations based on the read data, the initial equations representing the system and including a variable that represents a physical quantity to be determined; a boundary condition decision unit that reads a value that represents the physical quantity as data indicating a boundary condition, and decides a boundary condition; and a calculation unit that transforms the n initial equations into equations having 2n variables or equation including 2n equations; decides a known part that includes variables that are made known by the boundary condition and an unknown part that includes unknown variables, in the transformed equations having the 2n variables or the transformed equation including the 2n equations; and calculates a solution of the equations with regard to the unknown part.
 18. The information processing device according to claim 17, wherein the initial equation decision unit decides n differential equations having the variable that represents the physical quantity, and the calculation unit generates data indicating the 2n equations, using differential operators of the differential equations decided by the initial equation decision unit and adjoint differential operators decided according to the differential operators, and calculates solutions of the 2n equations, thereby outputting the physical quantity, the physical quantity being at least one.
 19. The information processing device according to claim 17, wherein the initial equation decision unit decides n equations as the initial equations, the n-equations including two n-dimensional variable vectors that indicate physical quantities at nodes of the constituent element of the system, and an n-row matrix, the boundary condition decision unit is able to decide a boundary condition in which, regarding the variable vector, the number of degrees of freedom of variables whose values are known, and the number of degrees of freedom of variables whose values are unknown, are different, and the calculation unit generates a 2n-dimensional vector based on the two variable vectors; transforms the n-row matrix into a 2n-column matrix based on variables of the 2n-dimensional vector; decides a known part and an unknown part among the variables of the 2n-dimensional vector, the known part including variables that are made known by the boundary condition, the unknown part containing unknown variables, degrees of freedom of the known part and the unknown part being not necessarily identical to each other; transforms the 2n-column matrix and the 2n-dimensional vector into a form such that the variables of the unknown part are expressed by the variables of the known part; and calculates the variables of the unknown part by using the transformed matrix.
 20. An information processing device that, in the case where there are a plurality of solutions calculated by the information processing device according to claim 17, decides a mode coefficient with respect to homogeneous solutions by using a functional Π of the following equation so that variation of the functional Π is zero, and determines a solution by using the decided mode coefficient: $\Pi \equiv {\sum\limits_{i}\; {\int_{S}{\left( {{\sum\limits_{j}\; {L_{ij}u_{j}}} - f_{i}} \right)^{2}{s}}}}$ where S represents an internal region of the system, L_(ij) represents a differential operator of a differential equation that the system should satisfy, and f_(i) and u_(i) represent variables representing physical quantities.
 21. The information processing device according to claim 20, that receives input of a value in the vicinities of the decided mode coefficient from a user, calculates a solution using the mode coefficient having the input value, and outputs the solution or information obtained from the solution.
 22. An information processing device comprising: an initial equation decision unit that reads data indicating a structure of a system as an object of processing and properties of a constituent element of the system, and decides differential equations as initial equations based on the read data, the differential equations representing the system and including a primal variable that represents a physical quantity to be determined; a boundary condition decision unit that reads a value that represents the physical amount as data indicating a boundary condition, and decides a boundary condition; and an adjoint boundary condition decision unit that, in the case where dual variables the number of which is the same as that of the variables of the differential equations and dual differential equations are defined, calculates a boundary term obtained by partial integration of a sum of integration, that is, an inner product, of a result of the differential operators of the differential equations acting on the variables with the dual variables, and decides an adjoint boundary condition that is a condition of dual variables that makes the boundary term zero under the boundary condition; and a determination unit that outputs a result of comparison between the adjoint boundary condition and the boundary condition, wherein the inner product of the result of the differential operators of the differential equations acting on the variables with the dual variables is equal to an inner product of the variables with a result of the differential operators of the dual differential equations acting on the dual variables.
 23. The information processing device according to claim 22, wherein whether the boundary condition and the adjoint boundary condition coincide or not is determined by using the following equation, according to whether combinations of respective known parts Fb and Ub of a node force F and a node displacement U, and combinations of respective known parts Fb* and Ub* of a dual node force F* and a dual node displacement U* coincide with each other: $\begin{matrix} {\mspace{79mu} {{\frac{R}{G} \equiv {\left\{ {\underset{1 \times n_{v_{b}}}{\left\{ U_{b} \right\}^{T}\mspace{14mu}}\text{?}\mspace{14mu} \underset{1 \times n_{F_{b}}}{\left\{ F_{b} \right\}^{T}\mspace{14mu}}\text{?}} \right\} \begin{Bmatrix} \text{?} \\ \underset{n_{F_{b}^{*}} \times 1}{\left\{ {- F_{b}^{*}} \right\}} \\ \text{?} \\ \underset{n_{v_{b}^{*}} \times 1}{\left\{ U_{b}^{*} \right\}} \end{Bmatrix}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (1814) \end{matrix}$
 24. An information processing device comprising: a setting unit that decides n initial equations expressed by an n-row matrix and two n-dimensional variables that represent physical quantities in a plurality of elements of a system as an object of processing; a boundary condition decision unit that reads a value that represents the physical quantities as data indicating a boundary condition, and decides a boundary condition; and a determination unit that decides a known part and an unknown part, the known part including a variable that is made known by the boundary condition among the two n-dimensional variables, an unknown part including an unknown variable among the same; and in the case where a degree of freedom of the unknown part of one variable is not equal to a degree of freedom of the known part of the other variable, outputs information notifying that the boundary condition is non-self-adjoint.
 25. An information processing device comprising: a setting unit that sets an original differential operator of an analysis object and a boundary condition of variables; an adjoint boundary condition calculation unit that calculates an adjoint boundary condition from the boundary condition; and a calculation unit that calculates a solution u_(j) of the analysis object by solving a simultaneous equations obtained by the following equation, in which, the solution u_(j) of the analysis object is expressed by a sum (u_(Bj)+u_(Hj)) of a term u_(Bj) that satisfies a inhomogeneous boundary condition and a term u_(Hj) that satisfies a homogeneous boundary condition, a function group that satisfies the primal boundary condition decided by the boundary condition and the adjoint boundary condition is substituted into u_(Hj), and a function group that satisfies the dual boundary condition decided by the boundary condition and the adjoint boundary condition is substituted into δu_(i)*, ${\sum\limits_{i}\; {\int_{S}{{\left( {{\sum\limits_{j}\; {L_{ij}u_{j}}} - f_{i}} \right) \cdot \delta}\; u_{i}^{*}\ {s}}}} = 0.$
 26. The information processing device according to claim 25, comprising: a setting unit that sets an original differential operator of an analysis object and a boundary condition of variables; an adjoint boundary condition calculation unit that calculates an adjoint boundary condition from the boundary condition; and a calculation unit that calculates a primal differential operator and a dual differential operator from the original differential operator, and determines a primal eigenfunction and a dual eigenfunction by using primal simultaneous differential equations and dual simultaneous differential equations, as well as the boundary condition and the adjoint boundary condition, thereby calculating a solution of simultaneous differential equations.
 27. The information processing device according to claim 26, further comprising: a self-adjoint determination unit that determines whether the boundary condition and the adjoint boundary condition coincide with each other; wherein, the calculation unit includes a self-adjoint calculation unit that, in the case where it is determined that the boundary condition and the adjoint boundary condition coincide with each other, calculates a solution of a self-adjoint problem by determining a self-adjoint eigenfunction of the self-adjoint problem from an original differential operator, and a non-self-adjoint calculation unit that, in the case where it is determined that the boundary condition and the adjoint boundary condition do not coincide with each other, determines a primal eigenfunction and a dual eigenfunction by using the primal simultaneous differential equations and the dual simultaneous differential equations, as well as the boundary condition and the adjoint boundary condition, thereby calculating a solution of simultaneous differential equations.
 28. The information processing device according to claim 26, wherein the solution u_(j) of the analysis object is expressed by a sum (u_(Bj)+u_(Bj)) of a term u_(Bj) that satisfies a inhomogeneous boundary condition and a term u_(Hj) that satisfies a homogeneous boundary condition, and in the case where the primal eigenfunction is given as φ_(j), and the dual eigenfunction is given as φ_(j)*, the primal variable which is to be determined as a primal solution of the primal simultaneous differential equations, the dual variable which is to be determined as a dual solution of the dual simultaneous differential equations, the self-adjoint differential equation, the primal simultaneous differential equations, and the dual simultaneous differential equations are expressed by the following equations: Primal variable: $u_{Hj} \equiv {\sum\limits_{k}\; {c_{k}\varphi_{jk}}}$ Dual variable: $u_{Hj}^{*} \equiv {\sum\limits_{k}\; {c_{k}^{*}\varphi_{jk}^{*}}}$ Self-adjoint differential equation: ${\sum\limits_{j}\; {L_{ij}\varphi_{j}}} = {\lambda \; w_{i}\varphi_{i}}$ Primal simultaneous differential equations: ${\sum\limits_{j}\; {L_{ij}\varphi_{j}}} = {\lambda \; w_{i}\varphi_{i}^{*}}$ Dual simultaneous differential equations: ${\sum\limits_{j}\; {L_{ij}^{*}\varphi_{j}^{*}}} = {\lambda \; w_{i}{\varphi_{i}^{*}.}}$
 29. The information processing device according to claim 17, wherein the boundary condition decision unit or the setting unit receives input of information indicating a part whose value is unknown of a variable that represents the physical quantity, from a user, and decides a boundary condition using this information.
 30. An information processing method comprising: an initial equation deciding step wherein a computer reads data indicating a structure of a system as an object of processing and properties of a constituent element of the system, and decides n initial equations based on the read data, the initial equations representing the system and including a variable that represents a physical quantity to be determined; a boundary condition deciding step wherein a computer reads a value that represents the physical quantity as data indicating a boundary condition, and decides a boundary condition; and a calculating step wherein a computer transforms the n initial equations into equations having 2n variables or equation including 2n equations; decides a known part that includes variables that are made known by the boundary condition and an unknown part that includes unknown variables, in the transformed equations having the 2n variables or the transformed equation including the 2n equations; and calculates a solution of the equations with regard to the unknown part.
 31. A non-transitory recording medium storing an information processing program that causes a computer to execute: initial equation deciding processing of reading data indicating a structure of a system as an object of processing and properties of a constituent element of the system, and deciding n initial equations based on the read data, the initial equations representing the system and including a variable that represents a physical quantity to be determined; boundary condition deciding processing of reading a value that represents the physical quantity as data indicating a boundary condition, and deciding a boundary condition; and calculating processing of transforming the n equations into equations having 2n variables or equation including 2n equations; deciding a known part that includes variables that are made known by the boundary condition and an unknown part that includes unknown variables, in the transformed equations having the 2n variables or the transformed equation including the 2n equations; and calculating a solution of the equations with regard to the unknown part.
 32. An information processing method comprising: an initial equation deciding step wherein a computer reads data indicating a structure of a system as an object of processing and properties of a constituent element of the system, and decides n differential equations as initial equations based on the read data, the differential equations representing the system and including a variable that represents a physical quantity to be determined; a boundary condition deciding step wherein a computer reads a value that represents the physical quantity as data indicating a boundary condition, and decides a boundary condition; and an adjoint boundary condition deciding step wherein, in the case where dual variables the number of which is the same as that of the variables of the differential equations and dual differential equations are defined, a computer calculates a boundary term obtained by partial integration of a sum of integration, that is, an inner product, of a result of the differential operators of the differential equations acting on the variables with the dual variables, and decides an adjoint boundary condition that is a condition of dual variables that makes the boundary term zero under the boundary condition; and an output step wherein a computer outputs a result of comparison between the adjoint boundary condition and the boundary condition, wherein the inner product of the result of the differential operators of the differential equations acting on the variables with the dual variables is equal to an inner product of the variables with a result of the differential operators of the dual differential equations acting on the dual variables.
 33. A non-transitory recording medium storing an information processing program that causes a computer to execute: initial equation deciding processing of reading data indicating a structure of a system as an object of processing and properties of a constituent element of the system, and deciding n differential equations as initial equations based on the read data, the differential equations representing the system and including a variable that represents a physical quantity to be determined; boundary condition deciding processing of reading a value that represents the physical quantity as data indicating a boundary condition, and deciding a boundary condition; and adjoint boundary condition deciding processing of, in the case where dual variables the number of which is the same as that of the variables of the differential equations and the dual simultaneous differential equations are defined, calculating a boundary term obtained by partial integration of a sum of integration, that is, an inner product, of a result of the differential operators of the differential equations acting on the variables with the dual variables, and deciding an adjoint boundary condition that is a condition of dual variables that makes the boundary term zero under the boundary condition; and output processing of outputting a result of comparison between the adjoint boundary condition and the boundary condition, wherein the inner product of the result of the differential operators of the differential equations acting on the variables with the dual variables is equal to an inner product of the variables with a result of the differential operators of the dual differential equations acting on the dual variables. 